L(s) = 1 | + 2.15·3-s + 1.53·5-s − 0.745·7-s + 1.62·9-s − 4.48·11-s + 2.05·13-s + 3.30·15-s − 17-s − 5.76·19-s − 1.60·21-s − 4.38·23-s − 2.63·25-s − 2.95·27-s + 8.97·29-s − 0.172·31-s − 9.64·33-s − 1.14·35-s + 6.69·37-s + 4.41·39-s + 1.53·41-s − 3.47·43-s + 2.49·45-s − 10.8·47-s − 6.44·49-s − 2.15·51-s + 3.84·53-s − 6.90·55-s + ⋯ |
L(s) = 1 | + 1.24·3-s + 0.688·5-s − 0.281·7-s + 0.541·9-s − 1.35·11-s + 0.569·13-s + 0.854·15-s − 0.242·17-s − 1.32·19-s − 0.349·21-s − 0.914·23-s − 0.526·25-s − 0.569·27-s + 1.66·29-s − 0.0309·31-s − 1.67·33-s − 0.193·35-s + 1.10·37-s + 0.706·39-s + 0.239·41-s − 0.529·43-s + 0.372·45-s − 1.57·47-s − 0.920·49-s − 0.301·51-s + 0.528·53-s − 0.930·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
good | 3 | \( 1 - 2.15T + 3T^{2} \) |
| 5 | \( 1 - 1.53T + 5T^{2} \) |
| 7 | \( 1 + 0.745T + 7T^{2} \) |
| 11 | \( 1 + 4.48T + 11T^{2} \) |
| 13 | \( 1 - 2.05T + 13T^{2} \) |
| 19 | \( 1 + 5.76T + 19T^{2} \) |
| 23 | \( 1 + 4.38T + 23T^{2} \) |
| 29 | \( 1 - 8.97T + 29T^{2} \) |
| 31 | \( 1 + 0.172T + 31T^{2} \) |
| 37 | \( 1 - 6.69T + 37T^{2} \) |
| 41 | \( 1 - 1.53T + 41T^{2} \) |
| 43 | \( 1 + 3.47T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 - 3.84T + 53T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 + 1.23T + 67T^{2} \) |
| 71 | \( 1 + 3.83T + 71T^{2} \) |
| 73 | \( 1 + 7.18T + 73T^{2} \) |
| 79 | \( 1 + 9.93T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−7.84624700820526265269789872630, −6.72748298746184036532429612137, −6.19014445739452968249731845734, −5.45787572797351813391755789438, −4.52717911907037210812964702375, −3.79972872103330323188742106399, −2.83240342867198430044281506404, −2.44152745290629393065288378378, −1.61544877776204764142287168069, 0,
1.61544877776204764142287168069, 2.44152745290629393065288378378, 2.83240342867198430044281506404, 3.79972872103330323188742106399, 4.52717911907037210812964702375, 5.45787572797351813391755789438, 6.19014445739452968249731845734, 6.72748298746184036532429612137, 7.84624700820526265269789872630