| L(s) = 1 | − 2-s + 4-s − 1.88·5-s − 8-s + 1.88·10-s + 0.732·11-s + 7.01·13-s + 16-s + 3.25·17-s − 5.13·19-s − 1.88·20-s − 0.732·22-s − 1.26·23-s − 1.46·25-s − 7.01·26-s − 29-s − 1.37·31-s − 32-s − 3.25·34-s − 5.19·37-s + 5.13·38-s + 1.88·40-s − 6.51·41-s + 10.9·43-s + 0.732·44-s + 1.26·46-s − 5.13·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.840·5-s − 0.353·8-s + 0.594·10-s + 0.220·11-s + 1.94·13-s + 0.250·16-s + 0.789·17-s − 1.17·19-s − 0.420·20-s − 0.156·22-s − 0.264·23-s − 0.292·25-s − 1.37·26-s − 0.185·29-s − 0.247·31-s − 0.176·32-s − 0.558·34-s − 0.854·37-s + 0.833·38-s + 0.297·40-s − 1.01·41-s + 1.66·43-s + 0.110·44-s + 0.186·46-s − 0.749·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 1.88T + 5T^{2} \) |
| 11 | \( 1 - 0.732T + 11T^{2} \) |
| 13 | \( 1 - 7.01T + 13T^{2} \) |
| 17 | \( 1 - 3.25T + 17T^{2} \) |
| 19 | \( 1 + 5.13T + 19T^{2} \) |
| 23 | \( 1 + 1.26T + 23T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 + 1.37T + 31T^{2} \) |
| 37 | \( 1 + 5.19T + 37T^{2} \) |
| 41 | \( 1 + 6.51T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 + 5.13T + 47T^{2} \) |
| 53 | \( 1 + 8.39T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 + 0.503T + 61T^{2} \) |
| 67 | \( 1 - 9.66T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 - 1.88T + 73T^{2} \) |
| 79 | \( 1 - 3.66T + 79T^{2} \) |
| 83 | \( 1 - 8.89T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−7.86125141364713702483630076126, −6.82026205669427947528884212094, −6.26065918921078081179565083060, −5.63534921917094030783097417830, −4.50953340919533019406736802243, −3.71318964380789984331453486367, −3.29298737397199806871255144878, −1.98062175652241331152461973520, −1.16637233150690820848915824995, 0,
1.16637233150690820848915824995, 1.98062175652241331152461973520, 3.29298737397199806871255144878, 3.71318964380789984331453486367, 4.50953340919533019406736802243, 5.63534921917094030783097417830, 6.26065918921078081179565083060, 6.82026205669427947528884212094, 7.86125141364713702483630076126
