L(s) = 1 | + 0.352·3-s − 5-s + 3.95·7-s − 2.87·9-s − 5.63·11-s − 2.53·13-s − 0.352·15-s + 1.26·17-s + 6.83·19-s + 1.39·21-s − 23-s + 25-s − 2.07·27-s − 7.69·29-s + 9.04·31-s − 1.98·33-s − 3.95·35-s + 2.90·37-s − 0.893·39-s + 0.188·41-s + 2.97·43-s + 2.87·45-s + 4.82·47-s + 8.61·49-s + 0.445·51-s − 0.328·53-s + 5.63·55-s + ⋯ |
L(s) = 1 | + 0.203·3-s − 0.447·5-s + 1.49·7-s − 0.958·9-s − 1.70·11-s − 0.702·13-s − 0.0909·15-s + 0.306·17-s + 1.56·19-s + 0.303·21-s − 0.208·23-s + 0.200·25-s − 0.398·27-s − 1.42·29-s + 1.62·31-s − 0.345·33-s − 0.667·35-s + 0.477·37-s − 0.143·39-s + 0.0294·41-s + 0.453·43-s + 0.428·45-s + 0.704·47-s + 1.23·49-s + 0.0623·51-s − 0.0451·53-s + 0.760·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 - 0.352T + 3T^{2} \) |
| 7 | \( 1 - 3.95T + 7T^{2} \) |
| 11 | \( 1 + 5.63T + 11T^{2} \) |
| 13 | \( 1 + 2.53T + 13T^{2} \) |
| 17 | \( 1 - 1.26T + 17T^{2} \) |
| 19 | \( 1 - 6.83T + 19T^{2} \) |
| 29 | \( 1 + 7.69T + 29T^{2} \) |
| 31 | \( 1 - 9.04T + 31T^{2} \) |
| 37 | \( 1 - 2.90T + 37T^{2} \) |
| 41 | \( 1 - 0.188T + 41T^{2} \) |
| 43 | \( 1 - 2.97T + 43T^{2} \) |
| 47 | \( 1 - 4.82T + 47T^{2} \) |
| 53 | \( 1 + 0.328T + 53T^{2} \) |
| 59 | \( 1 + 0.966T + 59T^{2} \) |
| 61 | \( 1 + 1.31T + 61T^{2} \) |
| 67 | \( 1 - 3.94T + 67T^{2} \) |
| 71 | \( 1 + 7.69T + 71T^{2} \) |
| 73 | \( 1 - 0.617T + 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 + 1.30T + 83T^{2} \) |
| 89 | \( 1 + 17.3T + 89T^{2} \) |
| 97 | \( 1 - 8.26T + 97T^{2} \) |
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
−7.63230566010342077606603631037, −7.30623105759480132141423192966, −5.88051121263310972857896877746, −5.32534425785212165440162848781, −4.90590623484730624439035236962, −4.00786121723702779469426664202, −2.86616207121364453900626204213, −2.50806471678826617676443481298, −1.27722080337919496522568413780, 0,
1.27722080337919496522568413780, 2.50806471678826617676443481298, 2.86616207121364453900626204213, 4.00786121723702779469426664202, 4.90590623484730624439035236962, 5.32534425785212165440162848781, 5.88051121263310972857896877746, 7.30623105759480132141423192966, 7.63230566010342077606603631037