| L(s) = 1 | + 2-s − 3·3-s + 8·4-s + 14·5-s − 3·6-s + 10·7-s + 23·8-s + 14·10-s + 22·11-s − 24·12-s − 91·13-s + 10·14-s − 42·15-s + 23·16-s − 37·17-s − 30·19-s + 112·20-s − 30·21-s + 22·22-s + 162·23-s − 69·24-s − 103·25-s − 91·26-s + 27·27-s + 80·28-s + 113·29-s − 42·30-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 0.577·3-s + 4-s + 1.25·5-s − 0.204·6-s + 0.539·7-s + 1.01·8-s + 0.442·10-s + 0.603·11-s − 0.577·12-s − 1.94·13-s + 0.190·14-s − 0.722·15-s + 0.359·16-s − 0.527·17-s − 0.362·19-s + 1.25·20-s − 0.311·21-s + 0.213·22-s + 1.46·23-s − 0.586·24-s − 0.823·25-s − 0.686·26-s + 0.192·27-s + 0.539·28-s + 0.723·29-s − 0.255·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.294329580\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.294329580\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 13 | $C_2$ | \( 1 + 7 p T + p^{3} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T - 7 T^{2} - p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 7 T + p^{3} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 10 T - 243 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 p T - 7 p^{2} T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 37 T - 3544 T^{2} + 37 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T - 5959 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 162 T + 14077 T^{2} - 162 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 113 T - 11620 T^{2} - 113 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 196 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 13 T - 50484 T^{2} + 13 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 285 T + 12304 T^{2} + 285 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 246 T - 18991 T^{2} - 246 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 462 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 537 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 576 T + 126397 T^{2} + 576 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 635 T + 176244 T^{2} - 635 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 202 T - 259959 T^{2} + 202 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 1086 T + 821485 T^{2} - 1086 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 805 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 884 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 518 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 194 T - 667333 T^{2} + 194 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 1202 T + 532131 T^{2} - 1202 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−15.96335157320361747400818033815, −15.50660185736389799146505023790, −14.71065406476176879426368043299, −14.40501914791398660770265232756, −13.52441183341948312103113357095, −13.33110249511723568313264495032, −12.10452018916418862968736305427, −12.05486450592735556234436127103, −11.15827264771961786886357061358, −10.72958571480985644882449018400, −9.792001252726823010044286848286, −9.612213938687824173923577419352, −8.302137017093270503925324577102, −7.49195676063671437102792455199, −6.51087938510444781682605443790, −6.38530517900362351922164538031, −4.98122494267059200731253698901, −4.74113648188002584159266979691, −2.69121489734570904503322389083, −1.73942523488203418917004659384,
1.73942523488203418917004659384, 2.69121489734570904503322389083, 4.74113648188002584159266979691, 4.98122494267059200731253698901, 6.38530517900362351922164538031, 6.51087938510444781682605443790, 7.49195676063671437102792455199, 8.302137017093270503925324577102, 9.612213938687824173923577419352, 9.792001252726823010044286848286, 10.72958571480985644882449018400, 11.15827264771961786886357061358, 12.05486450592735556234436127103, 12.10452018916418862968736305427, 13.33110249511723568313264495032, 13.52441183341948312103113357095, 14.40501914791398660770265232756, 14.71065406476176879426368043299, 15.50660185736389799146505023790, 15.96335157320361747400818033815
