| L(s) = 1 | + 2-s + 4-s − 4·7-s + 8-s − 3·11-s − 13-s − 4·14-s + 16-s + 2·19-s − 3·22-s − 9·23-s − 26-s − 4·28-s + 6·29-s − 10·31-s + 32-s − 7·37-s + 2·38-s − 6·41-s − 10·43-s − 3·44-s − 9·46-s + 9·47-s + 9·49-s − 52-s + 6·53-s − 4·56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.51·7-s + 0.353·8-s − 0.904·11-s − 0.277·13-s − 1.06·14-s + 1/4·16-s + 0.458·19-s − 0.639·22-s − 1.87·23-s − 0.196·26-s − 0.755·28-s + 1.11·29-s − 1.79·31-s + 0.176·32-s − 1.15·37-s + 0.324·38-s − 0.937·41-s − 1.52·43-s − 0.452·44-s − 1.32·46-s + 1.31·47-s + 9/7·49-s − 0.138·52-s + 0.824·53-s − 0.534·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.363894422782119867360518242609, −8.306062831970305451990581355898, −7.36795621125105414818541685991, −6.63677624175871154046150423379, −5.81976321259770853901001341055, −5.07677677920990094266136886409, −3.83908000445159553037705885039, −3.15830693147456185364685172480, −2.10426053628138036411870728311, 0,
2.10426053628138036411870728311, 3.15830693147456185364685172480, 3.83908000445159553037705885039, 5.07677677920990094266136886409, 5.81976321259770853901001341055, 6.63677624175871154046150423379, 7.36795621125105414818541685991, 8.306062831970305451990581355898, 9.363894422782119867360518242609
