| L(s) = 1 | + 2.80·2-s + 3-s + 5.87·4-s + 4.11·5-s + 2.80·6-s + 10.8·8-s + 9-s + 11.5·10-s − 4.01·11-s + 5.87·12-s + 0.753·13-s + 4.11·15-s + 18.7·16-s − 2.86·17-s + 2.80·18-s − 8.05·19-s + 24.1·20-s − 11.2·22-s − 6.62·23-s + 10.8·24-s + 11.9·25-s + 2.11·26-s + 27-s − 3.55·29-s + 11.5·30-s − 1.91·31-s + 30.9·32-s + ⋯ |
| L(s) = 1 | + 1.98·2-s + 0.577·3-s + 2.93·4-s + 1.83·5-s + 1.14·6-s + 3.84·8-s + 0.333·9-s + 3.65·10-s − 1.21·11-s + 1.69·12-s + 0.209·13-s + 1.06·15-s + 4.69·16-s − 0.695·17-s + 0.661·18-s − 1.84·19-s + 5.40·20-s − 2.40·22-s − 1.38·23-s + 2.22·24-s + 2.38·25-s + 0.414·26-s + 0.192·27-s − 0.660·29-s + 2.10·30-s − 0.344·31-s + 5.47·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.53152955\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.53152955\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 2.80T + 2T^{2} \) |
| 5 | \( 1 - 4.11T + 5T^{2} \) |
| 11 | \( 1 + 4.01T + 11T^{2} \) |
| 13 | \( 1 - 0.753T + 13T^{2} \) |
| 17 | \( 1 + 2.86T + 17T^{2} \) |
| 19 | \( 1 + 8.05T + 19T^{2} \) |
| 23 | \( 1 + 6.62T + 23T^{2} \) |
| 29 | \( 1 + 3.55T + 29T^{2} \) |
| 31 | \( 1 + 1.91T + 31T^{2} \) |
| 37 | \( 1 - 5.80T + 37T^{2} \) |
| 43 | \( 1 + 2.59T + 43T^{2} \) |
| 47 | \( 1 + 1.45T + 47T^{2} \) |
| 53 | \( 1 - 7.12T + 53T^{2} \) |
| 59 | \( 1 - 2.95T + 59T^{2} \) |
| 61 | \( 1 + 5.68T + 61T^{2} \) |
| 67 | \( 1 - 4.99T + 67T^{2} \) |
| 71 | \( 1 - 5.96T + 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 - 8.86T + 79T^{2} \) |
| 83 | \( 1 + 1.22T + 83T^{2} \) |
| 89 | \( 1 - 6.89T + 89T^{2} \) |
| 97 | \( 1 + 1.83T + 97T^{2} \) |
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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
−7.86213973516666735210585009183, −6.98727029524183884108776397599, −6.24985629310039137280744381219, −5.93579101957392956957106681750, −5.19078686991904488174345448028, −4.50623632373997787899065162412, −3.74921348904166014916246848355, −2.62435508923111055157110840280, −2.28165866718398723951305143508, −1.71552293201010633308245277380,
1.71552293201010633308245277380, 2.28165866718398723951305143508, 2.62435508923111055157110840280, 3.74921348904166014916246848355, 4.50623632373997787899065162412, 5.19078686991904488174345448028, 5.93579101957392956957106681750, 6.24985629310039137280744381219, 6.98727029524183884108776397599, 7.86213973516666735210585009183
