L(s) = 1 | + 1.21·2-s − 0.525·4-s + 4.90·7-s − 3.06·8-s + 11-s + 4.14·13-s + 5.95·14-s − 2.67·16-s + 5.33·17-s − 5.18·19-s + 1.21·22-s − 4·23-s + 5.03·26-s − 2.57·28-s − 1.80·29-s + 2.62·31-s + 2.88·32-s + 6.47·34-s + 5.80·37-s − 6.29·38-s − 1.80·41-s + 4.90·43-s − 0.525·44-s − 4.85·46-s − 7.05·47-s + 17.0·49-s − 2.17·52-s + ⋯ |
L(s) = 1 | + 0.858·2-s − 0.262·4-s + 1.85·7-s − 1.08·8-s + 0.301·11-s + 1.15·13-s + 1.59·14-s − 0.668·16-s + 1.29·17-s − 1.18·19-s + 0.258·22-s − 0.834·23-s + 0.987·26-s − 0.486·28-s − 0.335·29-s + 0.470·31-s + 0.510·32-s + 1.11·34-s + 0.954·37-s − 1.02·38-s − 0.282·41-s + 0.747·43-s − 0.0792·44-s − 0.716·46-s − 1.02·47-s + 2.43·49-s − 0.302·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.146490612\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.146490612\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 - 1.21T + 2T^{2} \) |
| 7 | \( 1 - 4.90T + 7T^{2} \) |
| 13 | \( 1 - 4.14T + 13T^{2} \) |
| 17 | \( 1 - 5.33T + 17T^{2} \) |
| 19 | \( 1 + 5.18T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 1.80T + 29T^{2} \) |
| 31 | \( 1 - 2.62T + 31T^{2} \) |
| 37 | \( 1 - 5.80T + 37T^{2} \) |
| 41 | \( 1 + 1.80T + 41T^{2} \) |
| 43 | \( 1 - 4.90T + 43T^{2} \) |
| 47 | \( 1 + 7.05T + 47T^{2} \) |
| 53 | \( 1 - 7.18T + 53T^{2} \) |
| 59 | \( 1 + 1.67T + 59T^{2} \) |
| 61 | \( 1 - 0.755T + 61T^{2} \) |
| 67 | \( 1 + 4.85T + 67T^{2} \) |
| 71 | \( 1 + 0.428T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 + 6.42T + 79T^{2} \) |
| 83 | \( 1 - 2.90T + 83T^{2} \) |
| 89 | \( 1 + 0.622T + 89T^{2} \) |
| 97 | \( 1 + 2.75T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.611418426049659731156187801170, −8.319440568842977582502777915081, −7.52213522491542998239252820619, −6.23457926262712208796389110711, −5.72652633179915019160833579688, −4.85498820906525652532797636383, −4.22545949719542738448259963100, −3.52725488187448770204150681972, −2.18552927336778953146604969676, −1.08868185482885373761069613152,
1.08868185482885373761069613152, 2.18552927336778953146604969676, 3.52725488187448770204150681972, 4.22545949719542738448259963100, 4.85498820906525652532797636383, 5.72652633179915019160833579688, 6.23457926262712208796389110711, 7.52213522491542998239252820619, 8.319440568842977582502777915081, 8.611418426049659731156187801170