L(s) = 1 | − 2-s − 11·4-s + 10·5-s + 58·7-s + 15·8-s − 10·10-s + 86·11-s − 26·13-s − 58·14-s + 61·16-s + 28·17-s + 166·19-s − 110·20-s − 86·22-s − 60·23-s + 75·25-s + 26·26-s − 638·28-s − 120·29-s − 78·31-s − 89·32-s − 28·34-s + 580·35-s + 360·37-s − 166·38-s + 150·40-s + 72·41-s + ⋯ |
L(s) = 1 | − 0.353·2-s − 1.37·4-s + 0.894·5-s + 3.13·7-s + 0.662·8-s − 0.316·10-s + 2.35·11-s − 0.554·13-s − 1.10·14-s + 0.953·16-s + 0.399·17-s + 2.00·19-s − 1.22·20-s − 0.833·22-s − 0.543·23-s + 3/5·25-s + 0.196·26-s − 4.30·28-s − 0.768·29-s − 0.451·31-s − 0.491·32-s − 0.141·34-s + 2.80·35-s + 1.59·37-s − 0.708·38-s + 0.592·40-s + 0.274·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(4.988839813\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.988839813\) |
\(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
good | 2 | $D_{4}$ | \( 1 + T + 3 p^{2} T^{2} + p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 58 T + 1510 T^{2} - 58 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 86 T + 4494 T^{2} - 86 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 28 T + 5670 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 166 T + 18550 T^{2} - 166 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 60 T + 7826 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 120 T + 49046 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 78 T - 6370 T^{2} + 78 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 360 T + 128198 T^{2} - 360 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 72 T + 73790 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 44 T + 152698 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 362 T + 224070 T^{2} + 362 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 396 T + 297790 T^{2} - 396 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 18 T + 282166 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 442 T + p^{3} T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 1322 T + 851022 T^{2} - 1322 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 634 T + 397278 T^{2} + 634 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 60 T + 299126 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 180 T + 993566 T^{2} + 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 714 T + 1069046 T^{2} + 714 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 852 T + 1530214 T^{2} - 852 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 32 T - 1144706 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.25315807299306766542123320622, −9.997652496979122938487706513283, −9.433016468424054404808565732620, −9.331536882568370571352147385596, −8.657463433371568838333234825770, −8.574147451340513888495596366671, −7.78417762021928256267872438669, −7.73262769021696024084892858072, −7.11818787810582685898114032204, −6.44535348320882873428089469799, −5.53340061149538499873103431620, −5.51634287739402116780731933202, −4.86238366891280101024770365773, −4.58246920457428794291610742332, −3.99468111463163028280554831103, −3.57020118374432779017926284434, −2.28604223210613806671512840991, −1.73971999622285931377419795011, −1.04220099329015994799628565018, −1.00728599928993162454275894778,
1.00728599928993162454275894778, 1.04220099329015994799628565018, 1.73971999622285931377419795011, 2.28604223210613806671512840991, 3.57020118374432779017926284434, 3.99468111463163028280554831103, 4.58246920457428794291610742332, 4.86238366891280101024770365773, 5.51634287739402116780731933202, 5.53340061149538499873103431620, 6.44535348320882873428089469799, 7.11818787810582685898114032204, 7.73262769021696024084892858072, 7.78417762021928256267872438669, 8.574147451340513888495596366671, 8.657463433371568838333234825770, 9.331536882568370571352147385596, 9.433016468424054404808565732620, 9.997652496979122938487706513283, 10.25315807299306766542123320622