L(s) = 1 | + (−0.658 − 1.67i)5-s + (−0.988 − 0.149i)7-s + (0.955 + 0.294i)9-s + (1.82 − 0.563i)11-s + (1.36 + 0.930i)17-s + (−0.5 + 0.866i)19-s + (1.57 − 1.07i)23-s + (−1.64 + 1.52i)25-s + (0.400 + 1.75i)35-s + (−1.12 − 1.40i)43-s + (−0.134 − 1.79i)45-s + (−0.535 − 0.496i)47-s + (0.955 + 0.294i)49-s + (−2.14 − 2.69i)55-s + (−0.109 + 1.46i)61-s + ⋯ |
L(s) = 1 | + (−0.658 − 1.67i)5-s + (−0.988 − 0.149i)7-s + (0.955 + 0.294i)9-s + (1.82 − 0.563i)11-s + (1.36 + 0.930i)17-s + (−0.5 + 0.866i)19-s + (1.57 − 1.07i)23-s + (−1.64 + 1.52i)25-s + (0.400 + 1.75i)35-s + (−1.12 − 1.40i)43-s + (−0.134 − 1.79i)45-s + (−0.535 − 0.496i)47-s + (0.955 + 0.294i)49-s + (−2.14 − 2.69i)55-s + (−0.109 + 1.46i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.268498031\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.268498031\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (0.988 + 0.149i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
good | 3 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 5 | \( 1 + (0.658 + 1.67i)T + (-0.733 + 0.680i)T^{2} \) |
| 11 | \( 1 + (-1.82 + 0.563i)T + (0.826 - 0.563i)T^{2} \) |
| 13 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 17 | \( 1 + (-1.36 - 0.930i)T + (0.365 + 0.930i)T^{2} \) |
| 23 | \( 1 + (-1.57 + 1.07i)T + (0.365 - 0.930i)T^{2} \) |
| 29 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 41 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 43 | \( 1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2} \) |
| 47 | \( 1 + (0.535 + 0.496i)T + (0.0747 + 0.997i)T^{2} \) |
| 53 | \( 1 + (0.988 - 0.149i)T^{2} \) |
| 59 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 61 | \( 1 + (0.109 - 1.46i)T + (-0.988 - 0.149i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 73 | \( 1 + (-0.733 + 0.680i)T + (0.0747 - 0.997i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.367 + 1.61i)T + (-0.900 + 0.433i)T^{2} \) |
| 89 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.741557554733924528691259994299, −7.949288400729124599760319530705, −7.06007636105039331587917744519, −6.36012734227503642895875553281, −5.52651980942087389578618612203, −4.60268169647647655558314582621, −3.88540942904998175243511463391, −3.47255376264008178031271292999, −1.58483494639346233134762855115, −0.925553352201926599715407459906,
1.28018722407742017471674057945, 2.75754281777982667396991546772, 3.40759393153548350317282677274, 3.92793913247470936542927677968, 4.98846370241939527548111207745, 6.32798233960809538214362156504, 6.75755488998173268999095769602, 7.09854815937167332526067952178, 7.80361326786672052848235755467, 9.122884254551185064407551800371