L(s) = 1 | + 20·2-s + 1.64e3·4-s + 6.25e3·5-s + 5.79e4·7-s + 9.85e4·8-s + 1.25e5·10-s + 6.18e5·11-s + 3.41e6·13-s + 1.15e6·14-s + 6.29e5·16-s − 1.31e6·17-s + 5.32e6·19-s + 1.02e7·20-s + 1.23e7·22-s − 5.89e7·23-s + 2.92e7·25-s + 6.82e7·26-s + 9.49e7·28-s − 9.41e7·29-s + 2.44e8·31-s + 1.18e8·32-s − 2.63e7·34-s + 3.61e8·35-s + 2.10e7·37-s + 1.06e8·38-s + 6.16e8·40-s + 7.45e8·41-s + ⋯ |
L(s) = 1 | + 0.441·2-s + 0.800·4-s + 0.894·5-s + 1.30·7-s + 1.06·8-s + 0.395·10-s + 1.15·11-s + 2.55·13-s + 0.575·14-s + 0.150·16-s − 0.225·17-s + 0.493·19-s + 0.716·20-s + 0.511·22-s − 1.90·23-s + 3/5·25-s + 1.12·26-s + 1.04·28-s − 0.852·29-s + 1.53·31-s + 0.622·32-s − 0.0994·34-s + 1.16·35-s + 0.0497·37-s + 0.218·38-s + 0.951·40-s + 1.00·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(10.39862471\) |
\(L(\frac12)\) |
\(\approx\) |
\(10.39862471\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p^{5} T )^{2} \) |
good | 2 | $D_{4}$ | \( 1 - 5 p^{2} T - 155 p^{3} T^{2} - 5 p^{13} T^{3} + p^{22} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 57900 T + 660624350 p T^{2} - 57900 p^{11} T^{3} + p^{22} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 618176 T + 245194892966 T^{2} - 618176 p^{11} T^{3} + p^{22} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 3414260 T + 6398662197390 T^{2} - 3414260 p^{11} T^{3} + p^{22} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 1317940 T + 59308395866630 T^{2} + 1317940 p^{11} T^{3} + p^{22} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 280280 p T + 194538827137638 T^{2} - 280280 p^{12} T^{3} + p^{22} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2562780 p T + 2773540471931410 T^{2} + 2562780 p^{12} T^{3} + p^{22} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3246220 p T + 23426350431097358 T^{2} + 3246220 p^{12} T^{3} + p^{22} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 244543464 T + 34393316207729486 T^{2} - 244543464 p^{11} T^{3} + p^{22} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 21003220 T + 137126715218410590 T^{2} - 21003220 p^{11} T^{3} + p^{22} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 745743316 T + 929792912462405846 T^{2} - 745743316 p^{11} T^{3} + p^{22} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 629950100 T + 1840945003918927050 T^{2} - 629950100 p^{11} T^{3} + p^{22} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 1402061540 T + 5181805952108806370 T^{2} - 1402061540 p^{11} T^{3} + p^{22} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 1138320580 T - 2203723231625575330 T^{2} + 1138320580 p^{11} T^{3} + p^{22} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 7317515560 T + 55027608950440780118 T^{2} + 7317515560 p^{11} T^{3} + p^{22} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1516425676 T + 85869525433683691566 T^{2} + 1516425676 p^{11} T^{3} + p^{22} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 15734290140 T + \)\(30\!\cdots\!30\)\( T^{2} - 15734290140 p^{11} T^{3} + p^{22} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 32938471544 T + \)\(73\!\cdots\!26\)\( T^{2} + 32938471544 p^{11} T^{3} + p^{22} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 29982848860 T + \)\(78\!\cdots\!10\)\( T^{2} + 29982848860 p^{11} T^{3} + p^{22} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 3302823120 T + \)\(12\!\cdots\!58\)\( T^{2} + 3302823120 p^{11} T^{3} + p^{22} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 13299102420 T + \)\(18\!\cdots\!30\)\( T^{2} + 13299102420 p^{11} T^{3} + p^{22} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12674770860 T + \)\(43\!\cdots\!78\)\( T^{2} - 12674770860 p^{11} T^{3} + p^{22} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 3080703740 T + \)\(18\!\cdots\!70\)\( T^{2} + 3080703740 p^{11} T^{3} + p^{22} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.59106970106290456839370614740, −13.56710128098156843287499157000, −12.55223479424372090912921781289, −11.69758221571725753929748401754, −11.40418902321999187145402067319, −10.90304658380817722644706725155, −10.36095623571164980771795751486, −9.483558352005119969532702819207, −8.773033566273907711214682152651, −8.150800460180734116353941224084, −7.52842368174729526223788513249, −6.50372507392322184884428906988, −6.12918124445120266732680931286, −5.58109298499102332828255387665, −4.32419327711743112445260313694, −4.16813968501628161083361706858, −3.00267797686282288936505338766, −1.83037914699779268756640265577, −1.60277424795465373480873090605, −0.986321171235488763153926762507,
0.986321171235488763153926762507, 1.60277424795465373480873090605, 1.83037914699779268756640265577, 3.00267797686282288936505338766, 4.16813968501628161083361706858, 4.32419327711743112445260313694, 5.58109298499102332828255387665, 6.12918124445120266732680931286, 6.50372507392322184884428906988, 7.52842368174729526223788513249, 8.150800460180734116353941224084, 8.773033566273907711214682152651, 9.483558352005119969532702819207, 10.36095623571164980771795751486, 10.90304658380817722644706725155, 11.40418902321999187145402067319, 11.69758221571725753929748401754, 12.55223479424372090912921781289, 13.56710128098156843287499157000, 13.59106970106290456839370614740