| L(s) = 1 | + 2.73·5-s − 11-s − 13-s − 4.19·17-s + 1.46·19-s − 2·23-s + 2.46·25-s − 8.19·29-s − 6.19·31-s − 10.3·37-s + 11.4·41-s + 0.732·43-s + 6.92·47-s − 7·49-s − 8.92·53-s − 2.73·55-s − 9.46·59-s − 0.535·61-s − 2.73·65-s − 7.26·67-s + 5.46·71-s − 2.53·73-s − 12.7·79-s − 6.92·83-s − 11.4·85-s + 16.1·89-s + 4·95-s + ⋯ |
| L(s) = 1 | + 1.22·5-s − 0.301·11-s − 0.277·13-s − 1.01·17-s + 0.335·19-s − 0.417·23-s + 0.492·25-s − 1.52·29-s − 1.11·31-s − 1.70·37-s + 1.79·41-s + 0.111·43-s + 1.01·47-s − 49-s − 1.22·53-s − 0.368·55-s − 1.23·59-s − 0.0686·61-s − 0.338·65-s − 0.887·67-s + 0.648·71-s − 0.296·73-s − 1.43·79-s − 0.760·83-s − 1.24·85-s + 1.71·89-s + 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 2.73T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 17 | \( 1 + 4.19T + 17T^{2} \) |
| 19 | \( 1 - 1.46T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 + 8.19T + 29T^{2} \) |
| 31 | \( 1 + 6.19T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 - 0.732T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 + 8.92T + 53T^{2} \) |
| 59 | \( 1 + 9.46T + 59T^{2} \) |
| 61 | \( 1 + 0.535T + 61T^{2} \) |
| 67 | \( 1 + 7.26T + 67T^{2} \) |
| 71 | \( 1 - 5.46T + 71T^{2} \) |
| 73 | \( 1 + 2.53T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 + 6.92T + 83T^{2} \) |
| 89 | \( 1 - 16.1T + 89T^{2} \) |
| 97 | \( 1 - 6.39T + 97T^{2} \) |
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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
−7.70870459854261669451416505328, −7.20163406280271187549707709174, −6.25811503001509403095343172216, −5.74108545448913993792157597088, −5.07150983658226462618248385353, −4.18758407134291447706232941180, −3.18853292357634367175279488304, −2.19692764080272511934484802374, −1.64324630623377192260049891341, 0,
1.64324630623377192260049891341, 2.19692764080272511934484802374, 3.18853292357634367175279488304, 4.18758407134291447706232941180, 5.07150983658226462618248385353, 5.74108545448913993792157597088, 6.25811503001509403095343172216, 7.20163406280271187549707709174, 7.70870459854261669451416505328
