| L(s) = 1 | − 4·3-s + 2·5-s + 3·7-s + 6·9-s + 11-s + 2·13-s − 8·15-s − 5·17-s + 4·19-s − 12·21-s − 2·23-s + 3·25-s + 4·27-s − 29-s + 11·31-s − 4·33-s + 6·35-s + 2·37-s − 8·39-s + 5·41-s − 9·43-s + 12·45-s + 6·47-s + 49-s + 20·51-s − 3·53-s + 2·55-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 0.894·5-s + 1.13·7-s + 2·9-s + 0.301·11-s + 0.554·13-s − 2.06·15-s − 1.21·17-s + 0.917·19-s − 2.61·21-s − 0.417·23-s + 3/5·25-s + 0.769·27-s − 0.185·29-s + 1.97·31-s − 0.696·33-s + 1.01·35-s + 0.328·37-s − 1.28·39-s + 0.780·41-s − 1.37·43-s + 1.78·45-s + 0.875·47-s + 1/7·49-s + 2.80·51-s − 0.412·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8761600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8761600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.559837712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.559837712\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 37 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - T + 14 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 32 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + T - 16 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 11 T + 84 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 5 T + 80 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 9 T + 98 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 100 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 14 T + 134 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T + 120 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2 T + 110 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 6 T + 122 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 126 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 10 T + 158 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 17 T + 192 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
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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
−8.923186591537118809313254549255, −8.550321507020406708585737031006, −8.114231127398290595473978649916, −7.924569988846724329534204396606, −7.17840316579356273189709496005, −6.87704405371642264543056432179, −6.42203068168969266833174528570, −6.23564868802375355028190118736, −5.86377634033121066503727511830, −5.59893517197025094665397616864, −5.09580754805127199400329086103, −4.84888094902420698194815799410, −4.34197979808937897181535749971, −4.31866281560689558279266458657, −3.08232806004888739081764269836, −3.01976122843927009797290716671, −2.00158036348357936263561176825, −1.73500845903593051092087327928, −0.940313057257450326121928764373, −0.59313526556338870536875821355,
0.59313526556338870536875821355, 0.940313057257450326121928764373, 1.73500845903593051092087327928, 2.00158036348357936263561176825, 3.01976122843927009797290716671, 3.08232806004888739081764269836, 4.31866281560689558279266458657, 4.34197979808937897181535749971, 4.84888094902420698194815799410, 5.09580754805127199400329086103, 5.59893517197025094665397616864, 5.86377634033121066503727511830, 6.23564868802375355028190118736, 6.42203068168969266833174528570, 6.87704405371642264543056432179, 7.17840316579356273189709496005, 7.924569988846724329534204396606, 8.114231127398290595473978649916, 8.550321507020406708585737031006, 8.923186591537118809313254549255
