| L(s) = 1 | − 2-s − 1.79·3-s + 4-s + 1.79·6-s + 2.79·7-s − 8-s + 0.208·9-s − 0.791·11-s − 1.79·12-s − 2.79·14-s + 16-s − 3.79·17-s − 0.208·18-s − 5·19-s − 5·21-s + 0.791·22-s + 4.58·23-s + 1.79·24-s + 5.00·27-s + 2.79·28-s + 0.791·29-s − 0.417·31-s − 32-s + 1.41·33-s + 3.79·34-s + 0.208·36-s + 7.37·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.03·3-s + 0.5·4-s + 0.731·6-s + 1.05·7-s − 0.353·8-s + 0.0695·9-s − 0.238·11-s − 0.517·12-s − 0.746·14-s + 0.250·16-s − 0.919·17-s − 0.0491·18-s − 1.14·19-s − 1.09·21-s + 0.168·22-s + 0.955·23-s + 0.365·24-s + 0.962·27-s + 0.527·28-s + 0.146·29-s − 0.0749·31-s − 0.176·32-s + 0.246·33-s + 0.650·34-s + 0.0347·36-s + 1.21·37-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 | \( 1 + 1.79T + 3T^{2} \) |
| 7 | \( 1 - 2.79T + 7T^{2} \) |
| 11 | \( 1 + 0.791T + 11T^{2} \) |
| 17 | \( 1 + 3.79T + 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 - 4.58T + 23T^{2} \) |
| 29 | \( 1 - 0.791T + 29T^{2} \) |
| 31 | \( 1 + 0.417T + 31T^{2} \) |
| 37 | \( 1 - 7.37T + 37T^{2} \) |
| 41 | \( 1 - 6.16T + 41T^{2} \) |
| 43 | \( 1 + 11.9T + 43T^{2} \) |
| 47 | \( 1 + 3.79T + 47T^{2} \) |
| 53 | \( 1 - 4.58T + 53T^{2} \) |
| 59 | \( 1 + 4.58T + 59T^{2} \) |
| 61 | \( 1 - 0.582T + 61T^{2} \) |
| 67 | \( 1 - 11T + 67T^{2} \) |
| 71 | \( 1 - 1.58T + 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 + 7.95T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−7.47499817893312182886139565383, −6.62320034353097826184798246277, −6.29444947398350431220611022720, −5.30850265702922406960744545266, −4.86461831588198730833190808979, −4.07902846137264622019073784626, −2.80730080084752413125111040396, −2.01486819530899281210635306560, −1.03070042475445301902260551175, 0,
1.03070042475445301902260551175, 2.01486819530899281210635306560, 2.80730080084752413125111040396, 4.07902846137264622019073784626, 4.86461831588198730833190808979, 5.30850265702922406960744545266, 6.29444947398350431220611022720, 6.62320034353097826184798246277, 7.47499817893312182886139565383
