| L(s) = 1 | − 4·2-s + 6·3-s + 12·4-s − 24·6-s − 14·7-s − 32·8-s + 27·9-s − 13·11-s + 72·12-s − 65·13-s + 56·14-s + 80·16-s + 35·17-s − 108·18-s + 6·19-s − 84·21-s + 52·22-s − 76·23-s − 192·24-s + 260·26-s + 108·27-s − 168·28-s − 70·29-s + 303·31-s − 192·32-s − 78·33-s − 140·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.63·6-s − 0.755·7-s − 1.41·8-s + 9-s − 0.356·11-s + 1.73·12-s − 1.38·13-s + 1.06·14-s + 5/4·16-s + 0.499·17-s − 1.41·18-s + 0.0724·19-s − 0.872·21-s + 0.503·22-s − 0.689·23-s − 1.63·24-s + 1.96·26-s + 0.769·27-s − 1.13·28-s − 0.448·29-s + 1.75·31-s − 1.06·32-s − 0.411·33-s − 0.706·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.262664829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.262664829\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 11 | $D_{4}$ | \( 1 + 13 T + 164 T^{2} + 13 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5 p T + 5168 T^{2} + 5 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 35 T + 9850 T^{2} - 35 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 3566 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 76 T + 24649 T^{2} + 76 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 70 T - 22253 T^{2} + 70 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 303 T + 68704 T^{2} - 303 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 103 T + 40452 T^{2} - 103 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 313 T + 159794 T^{2} - 313 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 p T + 62755 T^{2} + 6 p^{4} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 478 T + 173318 T^{2} - 478 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 273 T + 234816 T^{2} - 273 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 69 T + 310056 T^{2} - 69 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 1213 T + 821522 T^{2} - 1213 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 45 T + 579170 T^{2} + 45 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1589 T + 1346770 T^{2} - 1589 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 118 T + 184274 T^{2} + 118 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 401 T - 93972 T^{2} - 401 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 141 T + 1114392 T^{2} - 141 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 908 T + 1453478 T^{2} - 908 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 1330 T + p^{3} T^{2} )^{2} \) |
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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
−9.668666490923620848861102999888, −9.588327849677746410122138217908, −8.844645915823740507520472771832, −8.572954912488632462303385592242, −8.048670094194848149254373924846, −7.88749144094143973321540160931, −7.24266172090252329097200154632, −7.22813517095403926561393069062, −6.38030050975714684755169668310, −6.37495636038743009544493833419, −5.34368200875711467087612143896, −5.24005200375005610521027296984, −4.18667563847512058590941154772, −3.94709399531071229569246542385, −3.15524843254046232599103576116, −2.72980629560621389762621608985, −2.32154499376070764052774952327, −1.92565686358510412857499736754, −0.822413085482733427584890076729, −0.57817451270768014249303048235,
0.57817451270768014249303048235, 0.822413085482733427584890076729, 1.92565686358510412857499736754, 2.32154499376070764052774952327, 2.72980629560621389762621608985, 3.15524843254046232599103576116, 3.94709399531071229569246542385, 4.18667563847512058590941154772, 5.24005200375005610521027296984, 5.34368200875711467087612143896, 6.37495636038743009544493833419, 6.38030050975714684755169668310, 7.22813517095403926561393069062, 7.24266172090252329097200154632, 7.88749144094143973321540160931, 8.048670094194848149254373924846, 8.572954912488632462303385592242, 8.844645915823740507520472771832, 9.588327849677746410122138217908, 9.668666490923620848861102999888
