| L(s) = 1 | + 1.89·2-s + 0.0586·3-s + 1.60·4-s + 0.111·6-s − 2.85·7-s − 0.751·8-s − 2.99·9-s + 1.30·11-s + 0.0941·12-s − 0.0326·13-s − 5.42·14-s − 4.63·16-s + 6.87·17-s − 5.68·18-s − 7.81·19-s − 0.167·21-s + 2.47·22-s + 1.89·23-s − 0.0440·24-s − 0.0620·26-s − 0.351·27-s − 4.58·28-s + 4.83·29-s + 1.83·31-s − 7.29·32-s + 0.0766·33-s + 13.0·34-s + ⋯ |
| L(s) = 1 | + 1.34·2-s + 0.0338·3-s + 0.802·4-s + 0.0454·6-s − 1.07·7-s − 0.265·8-s − 0.998·9-s + 0.393·11-s + 0.0271·12-s − 0.00906·13-s − 1.44·14-s − 1.15·16-s + 1.66·17-s − 1.34·18-s − 1.79·19-s − 0.0365·21-s + 0.528·22-s + 0.394·23-s − 0.00899·24-s − 0.0121·26-s − 0.0676·27-s − 0.865·28-s + 0.898·29-s + 0.328·31-s − 1.28·32-s + 0.0133·33-s + 2.23·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.871908794\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.871908794\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 - 1.89T + 2T^{2} \) |
| 3 | \( 1 - 0.0586T + 3T^{2} \) |
| 7 | \( 1 + 2.85T + 7T^{2} \) |
| 11 | \( 1 - 1.30T + 11T^{2} \) |
| 13 | \( 1 + 0.0326T + 13T^{2} \) |
| 17 | \( 1 - 6.87T + 17T^{2} \) |
| 19 | \( 1 + 7.81T + 19T^{2} \) |
| 23 | \( 1 - 1.89T + 23T^{2} \) |
| 29 | \( 1 - 4.83T + 29T^{2} \) |
| 31 | \( 1 - 1.83T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 0.425T + 41T^{2} \) |
| 43 | \( 1 - 4.54T + 43T^{2} \) |
| 47 | \( 1 + 3.55T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 - 6.28T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 2.71T + 73T^{2} \) |
| 79 | \( 1 + 7.47T + 79T^{2} \) |
| 83 | \( 1 + 4.99T + 83T^{2} \) |
| 89 | \( 1 + 18.1T + 89T^{2} \) |
| 97 | \( 1 + 17.0T + 97T^{2} \) |
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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
−8.188758181296250309410615095128, −6.93698082356808613390086869595, −6.46724279246457108748511766202, −5.80629345164498747019055325933, −5.34005269802137597689501557043, −4.27821769461225545701722477690, −3.75214472661169758684782559316, −2.92253423802255095482512489386, −2.44039343476055881785143385568, −0.69609156815159149759178558064,
0.69609156815159149759178558064, 2.44039343476055881785143385568, 2.92253423802255095482512489386, 3.75214472661169758684782559316, 4.27821769461225545701722477690, 5.34005269802137597689501557043, 5.80629345164498747019055325933, 6.46724279246457108748511766202, 6.93698082356808613390086869595, 8.188758181296250309410615095128
