| L(s) = 1 | + 0.792·2-s + 2.52·3-s − 1.37·4-s + 2·6-s + 3.46·7-s − 2.67·8-s + 3.37·9-s − 11-s − 3.46·12-s + 2.74·14-s + 0.627·16-s − 5.04·17-s + 2.67·18-s + 4·19-s + 8.74·21-s − 0.792·22-s − 2.52·23-s − 6.74·24-s + 0.939·27-s − 4.75·28-s − 2.74·29-s − 2.37·31-s + 5.84·32-s − 2.52·33-s − 4·34-s − 4.62·36-s − 11.0·37-s + ⋯ |
| L(s) = 1 | + 0.560·2-s + 1.45·3-s − 0.686·4-s + 0.816·6-s + 1.30·7-s − 0.944·8-s + 1.12·9-s − 0.301·11-s − 1.00·12-s + 0.733·14-s + 0.156·16-s − 1.22·17-s + 0.629·18-s + 0.917·19-s + 1.90·21-s − 0.168·22-s − 0.526·23-s − 1.37·24-s + 0.180·27-s − 0.898·28-s − 0.509·29-s − 0.426·31-s + 1.03·32-s − 0.439·33-s − 0.685·34-s − 0.771·36-s − 1.81·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.253862319\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.253862319\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 0.792T + 2T^{2} \) |
| 3 | \( 1 - 2.52T + 3T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 5.04T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 2.52T + 23T^{2} \) |
| 29 | \( 1 + 2.74T + 29T^{2} \) |
| 31 | \( 1 + 2.37T + 31T^{2} \) |
| 37 | \( 1 + 11.0T + 37T^{2} \) |
| 41 | \( 1 + 2.74T + 41T^{2} \) |
| 43 | \( 1 - 3.46T + 43T^{2} \) |
| 47 | \( 1 - 6.63T + 47T^{2} \) |
| 53 | \( 1 + 3.16T + 53T^{2} \) |
| 59 | \( 1 + 1.62T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 + 0.644T + 67T^{2} \) |
| 71 | \( 1 - 7.11T + 71T^{2} \) |
| 73 | \( 1 + 6.92T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + 6.63T + 83T^{2} \) |
| 89 | \( 1 + 4.37T + 89T^{2} \) |
| 97 | \( 1 + 4.10T + 97T^{2} \) |
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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
−12.10644754515997214693555271085, −11.00822369540688062463973798039, −9.713511936023293326348629713878, −8.805944704445035951743390195337, −8.272881230858673627386982910731, −7.27109929736044219636591542524, −5.48454886309197426410481783222, −4.48242608704744081604113038338, −3.46283031192327173225695970565, −2.07253231868315618277971331391,
2.07253231868315618277971331391, 3.46283031192327173225695970565, 4.48242608704744081604113038338, 5.48454886309197426410481783222, 7.27109929736044219636591542524, 8.272881230858673627386982910731, 8.805944704445035951743390195337, 9.713511936023293326348629713878, 11.00822369540688062463973798039, 12.10644754515997214693555271085
