| L(s) = 1 | + 348. i·5-s + 907.·7-s − 1.18e4i·11-s − 4.09e3·13-s + 6.74e4i·17-s − 1.16e5·19-s − 6.83e4i·23-s + 2.69e5·25-s + 7.74e5i·29-s + 8.07e5·31-s + 3.16e5i·35-s + 1.02e6·37-s + 8.50e5i·41-s − 3.69e6·43-s − 5.47e6i·47-s + ⋯ |
| L(s) = 1 | + 0.557i·5-s + 0.377·7-s − 0.812i·11-s − 0.143·13-s + 0.807i·17-s − 0.896·19-s − 0.244i·23-s + 0.688·25-s + 1.09i·29-s + 0.874·31-s + 0.210i·35-s + 0.546·37-s + 0.301i·41-s − 1.07·43-s − 1.12i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.248303394\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.248303394\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 907.T \) |
| good | 5 | \( 1 - 348. iT - 3.90e5T^{2} \) |
| 11 | \( 1 + 1.18e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 4.09e3T + 8.15e8T^{2} \) |
| 17 | \( 1 - 6.74e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 1.16e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + 6.83e4iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 7.74e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 8.07e5T + 8.52e11T^{2} \) |
| 37 | \( 1 - 1.02e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 8.50e5iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 3.69e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 5.47e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 5.41e5iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 8.95e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 6.50e5T + 1.91e14T^{2} \) |
| 67 | \( 1 + 3.39e7T + 4.06e14T^{2} \) |
| 71 | \( 1 - 1.63e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 1.67e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 1.38e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 1.98e6iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 6.05e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 3.52e7T + 7.83e15T^{2} \) |
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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
−10.83047060483565479366164936559, −10.25504580608190183388815868061, −8.842271170246342252620132106253, −8.157800184014339387278374292247, −6.91182972416550155492429170453, −6.06796535432910509446706925463, −4.83648723284473093393286655338, −3.61102101183806487925204913287, −2.50119040997186430547496988115, −1.17263121574121461282975275848,
0.27753035086624517822821297409, 1.53725599471069034070629966466, 2.70764530435457799444224461160, 4.31125736506723930615192609217, 4.97842196066054894075095450454, 6.28667992460836316699889637117, 7.40601472688550700243667468611, 8.345237530126528606777939103849, 9.327314933080286102750983212373, 10.19272908130299453048323505184
