| L(s) = 1 | − 2·2-s − 3·3-s − 2·5-s + 6·6-s + 8·8-s + 4·10-s + 8·11-s − 84·13-s + 6·15-s − 16·16-s + 2·17-s + 124·19-s − 16·22-s − 76·23-s − 24·24-s + 125·25-s + 168·26-s + 27·27-s + 508·29-s − 12·30-s + 72·31-s − 24·33-s − 4·34-s − 398·37-s − 248·38-s + 252·39-s − 16·40-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 0.178·5-s + 0.408·6-s + 0.353·8-s + 0.126·10-s + 0.219·11-s − 1.79·13-s + 0.103·15-s − 1/4·16-s + 0.0285·17-s + 1.49·19-s − 0.155·22-s − 0.689·23-s − 0.204·24-s + 25-s + 1.26·26-s + 0.192·27-s + 3.25·29-s − 0.0730·30-s + 0.417·31-s − 0.126·33-s − 0.0201·34-s − 1.76·37-s − 1.05·38-s + 1.03·39-s − 0.0632·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.313058339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.313058339\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 3 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - 121 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 8 T - 1267 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 42 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 4909 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 124 T + 8517 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 76 T - 6391 T^{2} + 76 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 254 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 72 T - 24607 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 398 T + 107751 T^{2} + 398 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 462 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 212 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 264 T - 34127 T^{2} - 264 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 162 T - 122633 T^{2} - 162 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 772 T + 390605 T^{2} - 772 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 30 T - 226081 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 764 T + 282933 T^{2} - 764 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 236 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 418 T - 214293 T^{2} + 418 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 552 T - 188335 T^{2} + 552 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 1036 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 30 T - 704069 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 1190 T + p^{3} T^{2} )^{2} \) |
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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
−11.52727463213481976172045704171, −11.07443766045306164027814323012, −10.44686246803069579045987689704, −10.25539281831255275006967577502, −9.551528381142923962546914405606, −9.500653316873998047545419363587, −8.511891307323461194799340052919, −8.509382046497376126155616272364, −7.53714887398862525827833954880, −7.34712297921360505520584870228, −6.87328990228092010242518390699, −6.14297698309777683964648137273, −5.59358524038180540168885129591, −4.89388587069008611216404283598, −4.61955876626403286174948777698, −3.85206920583382157080152735104, −2.74173493525254529146793396765, −2.47697824517564855148506211167, −0.986466579720215026231979625906, −0.66895836114772020108419299615,
0.66895836114772020108419299615, 0.986466579720215026231979625906, 2.47697824517564855148506211167, 2.74173493525254529146793396765, 3.85206920583382157080152735104, 4.61955876626403286174948777698, 4.89388587069008611216404283598, 5.59358524038180540168885129591, 6.14297698309777683964648137273, 6.87328990228092010242518390699, 7.34712297921360505520584870228, 7.53714887398862525827833954880, 8.509382046497376126155616272364, 8.511891307323461194799340052919, 9.500653316873998047545419363587, 9.551528381142923962546914405606, 10.25539281831255275006967577502, 10.44686246803069579045987689704, 11.07443766045306164027814323012, 11.52727463213481976172045704171
