| L(s) = 1 | + 2-s + 4-s − 3.23·5-s + 3.85·7-s + 8-s − 3.23·10-s + 1.76·13-s + 3.85·14-s + 16-s − 1.76·17-s − 5·19-s − 3.23·20-s − 5.38·23-s + 5.47·25-s + 1.76·26-s + 3.85·28-s − 4.09·29-s − 7·31-s + 32-s − 1.76·34-s − 12.4·35-s + 9·37-s − 5·38-s − 3.23·40-s + 5.61·41-s − 12.7·43-s − 5.38·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.44·5-s + 1.45·7-s + 0.353·8-s − 1.02·10-s + 0.489·13-s + 1.03·14-s + 0.250·16-s − 0.427·17-s − 1.14·19-s − 0.723·20-s − 1.12·23-s + 1.09·25-s + 0.345·26-s + 0.728·28-s − 0.759·29-s − 1.25·31-s + 0.176·32-s − 0.302·34-s − 2.10·35-s + 1.47·37-s − 0.811·38-s − 0.511·40-s + 0.877·41-s − 1.93·43-s − 0.793·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 3.23T + 5T^{2} \) |
| 7 | \( 1 - 3.85T + 7T^{2} \) |
| 13 | \( 1 - 1.76T + 13T^{2} \) |
| 17 | \( 1 + 1.76T + 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 + 5.38T + 23T^{2} \) |
| 29 | \( 1 + 4.09T + 29T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 - 9T + 37T^{2} \) |
| 41 | \( 1 - 5.61T + 41T^{2} \) |
| 43 | \( 1 + 12.7T + 43T^{2} \) |
| 47 | \( 1 + 6.23T + 47T^{2} \) |
| 53 | \( 1 - 1.09T + 53T^{2} \) |
| 59 | \( 1 - 6.32T + 59T^{2} \) |
| 61 | \( 1 - 5.47T + 61T^{2} \) |
| 67 | \( 1 - 5.09T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 + 0.236T + 73T^{2} \) |
| 79 | \( 1 + 15T + 79T^{2} \) |
| 83 | \( 1 + 8.23T + 83T^{2} \) |
| 89 | \( 1 - 0.236T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−7.64527960779479084950751308844, −7.07781054676123091530112812406, −6.16172998603710615448323038448, −5.39505235120684757140825555175, −4.50100912823716058526708662426, −4.17019587144441244750950767170, −3.50941210671260548643821284211, −2.31464574019986610474399250348, −1.49579074398179906507658048453, 0,
1.49579074398179906507658048453, 2.31464574019986610474399250348, 3.50941210671260548643821284211, 4.17019587144441244750950767170, 4.50100912823716058526708662426, 5.39505235120684757140825555175, 6.16172998603710615448323038448, 7.07781054676123091530112812406, 7.64527960779479084950751308844
