| L(s) = 1 | + 2-s − 0.666·3-s + 4-s − 0.666·6-s + 2.41·7-s + 8-s − 2.55·9-s + 1.46·11-s − 0.666·12-s + 2.41·14-s + 16-s − 6.74·17-s − 2.55·18-s − 7.41·19-s − 1.60·21-s + 1.46·22-s + 7.48·23-s − 0.666·24-s + 3.70·27-s + 2.41·28-s − 0.696·29-s − 1.36·31-s + 32-s − 0.974·33-s − 6.74·34-s − 2.55·36-s + 9.25·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.384·3-s + 0.5·4-s − 0.272·6-s + 0.912·7-s + 0.353·8-s − 0.851·9-s + 0.440·11-s − 0.192·12-s + 0.645·14-s + 0.250·16-s − 1.63·17-s − 0.602·18-s − 1.70·19-s − 0.351·21-s + 0.311·22-s + 1.56·23-s − 0.136·24-s + 0.712·27-s + 0.456·28-s − 0.129·29-s − 0.245·31-s + 0.176·32-s − 0.169·33-s − 1.15·34-s − 0.425·36-s + 1.52·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 0.666T + 3T^{2} \) |
| 7 | \( 1 - 2.41T + 7T^{2} \) |
| 11 | \( 1 - 1.46T + 11T^{2} \) |
| 17 | \( 1 + 6.74T + 17T^{2} \) |
| 19 | \( 1 + 7.41T + 19T^{2} \) |
| 23 | \( 1 - 7.48T + 23T^{2} \) |
| 29 | \( 1 + 0.696T + 29T^{2} \) |
| 31 | \( 1 + 1.36T + 31T^{2} \) |
| 37 | \( 1 - 9.25T + 37T^{2} \) |
| 41 | \( 1 + 4.24T + 41T^{2} \) |
| 43 | \( 1 + 2.38T + 43T^{2} \) |
| 47 | \( 1 + 2.77T + 47T^{2} \) |
| 53 | \( 1 + 7.04T + 53T^{2} \) |
| 59 | \( 1 - 0.0690T + 59T^{2} \) |
| 61 | \( 1 - 5.91T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 - 3.60T + 71T^{2} \) |
| 73 | \( 1 + 9.30T + 73T^{2} \) |
| 79 | \( 1 + 0.766T + 79T^{2} \) |
| 83 | \( 1 + 16.5T + 83T^{2} \) |
| 89 | \( 1 + 17.7T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−7.22531070205731020752548436885, −6.55677935725326082074142943301, −6.11991239020196225736087575323, −5.24115069308776878628715161734, −4.61241164661014012057176998839, −4.18947852178982117866399012615, −3.04245053740561184115838438649, −2.31457673784366329319405051080, −1.43424803943199670427315745746, 0,
1.43424803943199670427315745746, 2.31457673784366329319405051080, 3.04245053740561184115838438649, 4.18947852178982117866399012615, 4.61241164661014012057176998839, 5.24115069308776878628715161734, 6.11991239020196225736087575323, 6.55677935725326082074142943301, 7.22531070205731020752548436885
