L(s) = 1 | + (0.923 − 0.382i)7-s + (0.923 + 0.382i)9-s + (−0.324 − 1.63i)11-s + (−0.707 + 1.70i)23-s + (−0.382 − 0.923i)25-s + (0.216 − 1.08i)29-s + (−0.324 − 0.216i)37-s + (0.382 − 0.0761i)43-s + (0.707 − 0.707i)49-s + (0.382 + 1.92i)53-s + 63-s + (1.92 + 0.382i)67-s + (−1.30 + 0.541i)71-s + (−0.923 − 1.38i)77-s + (0.541 − 0.541i)79-s + ⋯ |
L(s) = 1 | + (0.923 − 0.382i)7-s + (0.923 + 0.382i)9-s + (−0.324 − 1.63i)11-s + (−0.707 + 1.70i)23-s + (−0.382 − 0.923i)25-s + (0.216 − 1.08i)29-s + (−0.324 − 0.216i)37-s + (0.382 − 0.0761i)43-s + (0.707 − 0.707i)49-s + (0.382 + 1.92i)53-s + 63-s + (1.92 + 0.382i)67-s + (−1.30 + 0.541i)71-s + (−0.923 − 1.38i)77-s + (0.541 − 0.541i)79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.302623369\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.302623369\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-0.923 + 0.382i)T \) |
good | 3 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 5 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 11 | \( 1 + (0.324 + 1.63i)T + (-0.923 + 0.382i)T^{2} \) |
| 13 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 23 | \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 29 | \( 1 + (-0.216 + 1.08i)T + (-0.923 - 0.382i)T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (0.324 + 0.216i)T + (0.382 + 0.923i)T^{2} \) |
| 41 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 43 | \( 1 + (-0.382 + 0.0761i)T + (0.923 - 0.382i)T^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (-0.382 - 1.92i)T + (-0.923 + 0.382i)T^{2} \) |
| 59 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 61 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 67 | \( 1 + (-1.92 - 0.382i)T + (0.923 + 0.382i)T^{2} \) |
| 71 | \( 1 + (1.30 - 0.541i)T + (0.707 - 0.707i)T^{2} \) |
| 73 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 79 | \( 1 + (-0.541 + 0.541i)T - iT^{2} \) |
| 83 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 89 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 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
−9.453778562674986194978372594865, −8.405694126926586462038111156814, −7.899678624446398701243590814499, −7.24236195955419635357085568224, −6.06308290251504574172390975411, −5.42562856790230035897691622382, −4.37590599866515529994552605244, −3.67337314230797719050101698519, −2.34785295187339635611399767298, −1.15533648557837094571871009355,
1.56911931094038941806226569205, 2.36996828376079843955614681352, 3.85622581559046011263019762048, 4.67662079254488920165577214778, 5.24954646892691534297332933697, 6.52983736992735873510218113109, 7.16637673956321214840601145867, 7.922596827693169632550445003478, 8.724758355715335702905957732041, 9.622477013637402721813738560690