L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s + 9-s + 10-s − 11-s − 12-s − 13-s − 14-s − 15-s + 16-s − 17-s + 18-s − 19-s + 20-s + 21-s − 22-s + 23-s − 24-s + 25-s − 26-s − 27-s − 28-s + ⋯ |
L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s + 9-s + 10-s − 11-s − 12-s − 13-s − 14-s − 15-s + 16-s − 17-s + 18-s − 19-s + 20-s + 21-s − 22-s + 23-s − 24-s + 25-s − 26-s − 27-s − 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.107354323\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.107354323\) |
\(L(1)\) |
\(\approx\) |
\(1.299093061\) |
\(L(1)\) |
\(\approx\) |
\(1.299093061\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 41 | \( 1 \) |
good | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 - T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−34.47546215947206763426136146568, −33.6661521839196955099062132230, −32.69373074221944522676079722491, −31.65407261004172014783956532469, −29.93039824682136545224343416482, −29.1356468523485230293113881357, −28.541212871826520343874760780792, −26.39663799101612925813311486865, −25.09402232783170755205407028706, −23.946754832260061054858339580998, −22.71883368285344400541134984112, −21.93785421040218099753139884926, −20.91174866242924281026996303425, −19.13793575707422622348777862662, −17.44547462637966107901631736435, −16.3928748224285834048766910693, −15.1131494584028990933990985924, −13.22815980831450135879449383492, −12.71210543720683323663551991250, −10.969009693594641705466479713, −9.86659424108116642926443752381, −6.97434799988083841026559542165, −5.94119908387877980482735297655, −4.721950906732446356931143772002, −2.47534046077300626035956606785,
2.47534046077300626035956606785, 4.721950906732446356931143772002, 5.94119908387877980482735297655, 6.97434799988083841026559542165, 9.86659424108116642926443752381, 10.969009693594641705466479713, 12.71210543720683323663551991250, 13.22815980831450135879449383492, 15.1131494584028990933990985924, 16.3928748224285834048766910693, 17.44547462637966107901631736435, 19.13793575707422622348777862662, 20.91174866242924281026996303425, 21.93785421040218099753139884926, 22.71883368285344400541134984112, 23.946754832260061054858339580998, 25.09402232783170755205407028706, 26.39663799101612925813311486865, 28.541212871826520343874760780792, 29.1356468523485230293113881357, 29.93039824682136545224343416482, 31.65407261004172014783956532469, 32.69373074221944522676079722491, 33.6661521839196955099062132230, 34.47546215947206763426136146568