| L(s) = 1 | − 524.·5-s + (423. − 802. i)7-s − 4.37e3i·11-s − 913. i·13-s − 3.44e4·17-s − 2.34e3i·19-s − 8.24e4i·23-s + 1.96e5·25-s − 2.26e5i·29-s − 1.42e5i·31-s + (−2.21e5 + 4.20e5i)35-s + 1.95e5·37-s + 3.77e5·41-s − 3.85e5·43-s + 1.39e5·47-s + ⋯ |
| L(s) = 1 | − 1.87·5-s + (0.466 − 0.884i)7-s − 0.991i·11-s − 0.115i·13-s − 1.70·17-s − 0.0784i·19-s − 1.41i·23-s + 2.51·25-s − 1.72i·29-s − 0.858i·31-s + (−0.874 + 1.65i)35-s + 0.635·37-s + 0.855·41-s − 0.739·43-s + 0.196·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.2294752047\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2294752047\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + (-423. + 802. i)T \) |
| good | 5 | \( 1 + 524.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 4.37e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 + 913. iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 3.44e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.34e3iT - 8.93e8T^{2} \) |
| 23 | \( 1 + 8.24e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 + 2.26e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 1.42e5iT - 2.75e10T^{2} \) |
| 37 | \( 1 - 1.95e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.77e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.85e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.39e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.17e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 1.12e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.34e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 + 3.98e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.47e6iT - 9.09e12T^{2} \) |
| 73 | \( 1 + 5.03e5iT - 1.10e13T^{2} \) |
| 79 | \( 1 - 2.17e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.61e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.89e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.26e7iT - 8.07e13T^{2} \) |
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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
−10.52188214093100618963089956257, −8.853610756580405001842778324827, −8.112225180139344368251681539883, −7.38392547790022344080859263414, −6.29199795793240571797876321015, −4.41477274177139218009217193164, −4.15273778258869236701836497078, −2.73608071941221802311330480808, −0.74039702956566711316416782765, −0.079482422362205239951931325900,
1.66587912471335502388619566248, 3.11685378132650504650180558273, 4.30428850204951432325551941047, 5.06337944965587269155087616793, 6.76614628298472115063137972867, 7.56165628172861858570162205067, 8.498174078304500130371471666739, 9.259593725824863707673615326742, 10.85467909924778245357848287143, 11.46406058447157839804725353773
