| L(s) = 1 | + 2-s + 3-s − 5-s + 6-s − 8-s − 10-s − 4·13-s − 15-s − 16-s + 6·17-s − 4·19-s − 24-s − 4·26-s − 27-s − 12·29-s − 30-s + 8·31-s + 6·34-s − 2·37-s − 4·38-s − 4·39-s + 40-s + 12·41-s − 8·43-s − 48-s + 6·51-s + 6·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 0.447·5-s + 0.408·6-s − 0.353·8-s − 0.316·10-s − 1.10·13-s − 0.258·15-s − 1/4·16-s + 1.45·17-s − 0.917·19-s − 0.204·24-s − 0.784·26-s − 0.192·27-s − 2.22·29-s − 0.182·30-s + 1.43·31-s + 1.02·34-s − 0.328·37-s − 0.648·38-s − 0.640·39-s + 0.158·40-s + 1.87·41-s − 1.21·43-s − 0.144·48-s + 0.840·51-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.300352486\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.300352486\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.999455828235689195457185090904, −9.257930923204582788890108078237, −8.957708445990459306798124837534, −8.345863822808380456985608810940, −8.202916362463630845185988018292, −7.58932423213480981116045275959, −7.34164395307515912174007795081, −7.08461696689027482813478522335, −6.20537445365979675710518545164, −6.08720726301409836765464791644, −5.42834896561593495817217369094, −5.19853337791433568670991538822, −4.40953861238664577893831957836, −4.35974068274400443492239123835, −3.65863340018501041038093033770, −3.28978818186339699449988687586, −2.77959941454233904484367971001, −2.24408576449534734034468652085, −1.59253655493677423458052276415, −0.51118839732914461250351631195,
0.51118839732914461250351631195, 1.59253655493677423458052276415, 2.24408576449534734034468652085, 2.77959941454233904484367971001, 3.28978818186339699449988687586, 3.65863340018501041038093033770, 4.35974068274400443492239123835, 4.40953861238664577893831957836, 5.19853337791433568670991538822, 5.42834896561593495817217369094, 6.08720726301409836765464791644, 6.20537445365979675710518545164, 7.08461696689027482813478522335, 7.34164395307515912174007795081, 7.58932423213480981116045275959, 8.202916362463630845185988018292, 8.345863822808380456985608810940, 8.957708445990459306798124837534, 9.257930923204582788890108078237, 9.999455828235689195457185090904
