| L(s) = 1 | + 3.20·5-s + 7-s − 2·11-s − 7.45·17-s − 19-s + 4.65·23-s + 5.25·25-s − 7.85·29-s − 4.40·31-s + 3.20·35-s − 1.74·37-s + 6·41-s − 4.65·43-s − 9.20·47-s + 49-s − 2.54·53-s − 6.40·55-s + 8.40·61-s + 4·67-s + 1.05·71-s − 8.90·73-s − 2·77-s + 5.05·79-s − 15.3·83-s − 23.8·85-s − 10·89-s − 3.20·95-s + ⋯ |
| L(s) = 1 | + 1.43·5-s + 0.377·7-s − 0.603·11-s − 1.80·17-s − 0.229·19-s + 0.970·23-s + 1.05·25-s − 1.45·29-s − 0.790·31-s + 0.541·35-s − 0.287·37-s + 0.937·41-s − 0.710·43-s − 1.34·47-s + 0.142·49-s − 0.349·53-s − 0.863·55-s + 1.07·61-s + 0.488·67-s + 0.124·71-s − 1.04·73-s − 0.227·77-s + 0.569·79-s − 1.68·83-s − 2.58·85-s − 1.05·89-s − 0.328·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 3.20T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 7.45T + 17T^{2} \) |
| 23 | \( 1 - 4.65T + 23T^{2} \) |
| 29 | \( 1 + 7.85T + 29T^{2} \) |
| 31 | \( 1 + 4.40T + 31T^{2} \) |
| 37 | \( 1 + 1.74T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 4.65T + 43T^{2} \) |
| 47 | \( 1 + 9.20T + 47T^{2} \) |
| 53 | \( 1 + 2.54T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 8.40T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 1.05T + 71T^{2} \) |
| 73 | \( 1 + 8.90T + 73T^{2} \) |
| 79 | \( 1 - 5.05T + 79T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−7.18390645794579284806234644875, −6.67219117774276579498880778718, −5.92968058880144730332389231850, −5.30585455806748514551997085663, −4.77530129073923451075319748379, −3.86469642565869036680454961500, −2.77245368127003171968651928436, −2.12416130657522951878565920036, −1.50274398551007104576105623354, 0,
1.50274398551007104576105623354, 2.12416130657522951878565920036, 2.77245368127003171968651928436, 3.86469642565869036680454961500, 4.77530129073923451075319748379, 5.30585455806748514551997085663, 5.92968058880144730332389231850, 6.67219117774276579498880778718, 7.18390645794579284806234644875
