| L(s) = 1 | − 2-s − 2.23·3-s + 4-s − 5-s + 2.23·6-s + 7-s − 8-s + 2.00·9-s + 10-s − 2.23·12-s + 13-s − 14-s + 2.23·15-s + 16-s − 1.23·17-s − 2.00·18-s − 2.23·19-s − 20-s − 2.23·21-s − 3.76·23-s + 2.23·24-s + 25-s − 26-s + 2.23·27-s + 28-s + 4.47·29-s − 2.23·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.29·3-s + 0.5·4-s − 0.447·5-s + 0.912·6-s + 0.377·7-s − 0.353·8-s + 0.666·9-s + 0.316·10-s − 0.645·12-s + 0.277·13-s − 0.267·14-s + 0.577·15-s + 0.250·16-s − 0.299·17-s − 0.471·18-s − 0.512·19-s − 0.223·20-s − 0.487·21-s − 0.784·23-s + 0.456·24-s + 0.200·25-s − 0.196·26-s + 0.430·27-s + 0.188·28-s + 0.830·29-s − 0.408·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2.23T + 3T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + 1.23T + 17T^{2} \) |
| 19 | \( 1 + 2.23T + 19T^{2} \) |
| 23 | \( 1 + 3.76T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 5.23T + 31T^{2} \) |
| 37 | \( 1 - 8.94T + 37T^{2} \) |
| 41 | \( 1 + 7.23T + 41T^{2} \) |
| 43 | \( 1 - 3.23T + 43T^{2} \) |
| 47 | \( 1 - 7.70T + 47T^{2} \) |
| 53 | \( 1 + 5.70T + 53T^{2} \) |
| 59 | \( 1 - 4.70T + 59T^{2} \) |
| 61 | \( 1 + 3.52T + 61T^{2} \) |
| 67 | \( 1 + 7.23T + 67T^{2} \) |
| 71 | \( 1 + 1.52T + 71T^{2} \) |
| 73 | \( 1 + 7.23T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 + 0.236T + 83T^{2} \) |
| 89 | \( 1 + 17.2T + 89T^{2} \) |
| 97 | \( 1 - 3.52T + 97T^{2} \) |
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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
−7.42491129862571556621323947336, −6.74017309806897451830798333828, −6.09974513025417322749803845763, −5.57561026053187447106754248224, −4.66475354902551757530558363009, −4.09190823856226042749472426122, −2.96948523505928727454648456026, −1.92814177470901809841386216252, −0.919957714013372119031431770166, 0,
0.919957714013372119031431770166, 1.92814177470901809841386216252, 2.96948523505928727454648456026, 4.09190823856226042749472426122, 4.66475354902551757530558363009, 5.57561026053187447106754248224, 6.09974513025417322749803845763, 6.74017309806897451830798333828, 7.42491129862571556621323947336
