| L(s) = 1 | − 2.60·2-s + 4.79·4-s + 1.35·5-s − 7.27·8-s − 3.52·10-s + 2.74·11-s − 2.69·13-s + 9.37·16-s + 17-s − 4.43·19-s + 6.47·20-s − 7.14·22-s + 2.62·23-s − 3.17·25-s + 7.01·26-s − 4.71·29-s − 5.36·31-s − 9.87·32-s − 2.60·34-s + 4.93·37-s + 11.5·38-s − 9.83·40-s + 7.62·41-s − 10.3·43-s + 13.1·44-s − 6.82·46-s + 2.18·47-s + ⋯ |
| L(s) = 1 | − 1.84·2-s + 2.39·4-s + 0.604·5-s − 2.57·8-s − 1.11·10-s + 0.826·11-s − 0.747·13-s + 2.34·16-s + 0.242·17-s − 1.01·19-s + 1.44·20-s − 1.52·22-s + 0.546·23-s − 0.634·25-s + 1.37·26-s − 0.875·29-s − 0.963·31-s − 1.74·32-s − 0.446·34-s + 0.810·37-s + 1.87·38-s − 1.55·40-s + 1.19·41-s − 1.58·43-s + 1.97·44-s − 1.00·46-s + 0.318·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7497 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7497 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7700385088\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7700385088\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + 2.60T + 2T^{2} \) |
| 5 | \( 1 - 1.35T + 5T^{2} \) |
| 11 | \( 1 - 2.74T + 11T^{2} \) |
| 13 | \( 1 + 2.69T + 13T^{2} \) |
| 19 | \( 1 + 4.43T + 19T^{2} \) |
| 23 | \( 1 - 2.62T + 23T^{2} \) |
| 29 | \( 1 + 4.71T + 29T^{2} \) |
| 31 | \( 1 + 5.36T + 31T^{2} \) |
| 37 | \( 1 - 4.93T + 37T^{2} \) |
| 41 | \( 1 - 7.62T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 - 2.18T + 47T^{2} \) |
| 53 | \( 1 + 2.83T + 53T^{2} \) |
| 59 | \( 1 - 4.74T + 59T^{2} \) |
| 61 | \( 1 - 1.19T + 61T^{2} \) |
| 67 | \( 1 + 5.24T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 - 3.05T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 + 1.09T + 89T^{2} \) |
| 97 | \( 1 - 9.06T + 97T^{2} \) |
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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
−7.896793039023486892571983865651, −7.43579659608888178061149133754, −6.67820613598062659495076147907, −6.16783181335908703495546723274, −5.38263891191719599904102773464, −4.23309691197344867395758974001, −3.17761418111722696083406605855, −2.16648321279500888623858363468, −1.71376849641783375021264273576, −0.57693792998978917290326809069,
0.57693792998978917290326809069, 1.71376849641783375021264273576, 2.16648321279500888623858363468, 3.17761418111722696083406605855, 4.23309691197344867395758974001, 5.38263891191719599904102773464, 6.16783181335908703495546723274, 6.67820613598062659495076147907, 7.43579659608888178061149133754, 7.896793039023486892571983865651
