L(s) = 1 | + 2-s + 8·3-s − 7·4-s + 5·5-s + 8·6-s − 15·8-s + 37·9-s + 5·10-s + 12·11-s − 56·12-s + 78·13-s + 40·15-s + 41·16-s + 94·17-s + 37·18-s − 40·19-s − 35·20-s + 12·22-s + 32·23-s − 120·24-s + 25·25-s + 78·26-s + 80·27-s − 50·29-s + 40·30-s + 248·31-s + 161·32-s + ⋯ |
L(s) = 1 | + 0.353·2-s + 1.53·3-s − 7/8·4-s + 0.447·5-s + 0.544·6-s − 0.662·8-s + 1.37·9-s + 0.158·10-s + 0.328·11-s − 1.34·12-s + 1.66·13-s + 0.688·15-s + 0.640·16-s + 1.34·17-s + 0.484·18-s − 0.482·19-s − 0.391·20-s + 0.116·22-s + 0.290·23-s − 1.02·24-s + 1/5·25-s + 0.588·26-s + 0.570·27-s − 0.320·29-s + 0.243·30-s + 1.43·31-s + 0.889·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.361602356\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.361602356\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - p T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - T + p^{3} T^{2} \) |
| 3 | \( 1 - 8 T + p^{3} T^{2} \) |
| 11 | \( 1 - 12 T + p^{3} T^{2} \) |
| 13 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 17 | \( 1 - 94 T + p^{3} T^{2} \) |
| 19 | \( 1 + 40 T + p^{3} T^{2} \) |
| 23 | \( 1 - 32 T + p^{3} T^{2} \) |
| 29 | \( 1 + 50 T + p^{3} T^{2} \) |
| 31 | \( 1 - 8 p T + p^{3} T^{2} \) |
| 37 | \( 1 + 434 T + p^{3} T^{2} \) |
| 41 | \( 1 + 402 T + p^{3} T^{2} \) |
| 43 | \( 1 + 68 T + p^{3} T^{2} \) |
| 47 | \( 1 + 536 T + p^{3} T^{2} \) |
| 53 | \( 1 - 22 T + p^{3} T^{2} \) |
| 59 | \( 1 - 560 T + p^{3} T^{2} \) |
| 61 | \( 1 - 278 T + p^{3} T^{2} \) |
| 67 | \( 1 + 164 T + p^{3} T^{2} \) |
| 71 | \( 1 - 672 T + p^{3} T^{2} \) |
| 73 | \( 1 + 82 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1000 T + p^{3} T^{2} \) |
| 83 | \( 1 - 448 T + p^{3} T^{2} \) |
| 89 | \( 1 - 870 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1026 T + p^{3} T^{2} \) |
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
−11.91152298250087572989483998083, −10.32449259698026339247485307302, −9.544242905357488317009195169613, −8.554194504948999398194468424229, −8.244769374928595340441659614417, −6.59879768933006076589824751827, −5.30167294480111430917057451079, −3.85010069753357916697674857257, −3.18707343436852699114006984956, −1.43668475600251259916328283691,
1.43668475600251259916328283691, 3.18707343436852699114006984956, 3.85010069753357916697674857257, 5.30167294480111430917057451079, 6.59879768933006076589824751827, 8.244769374928595340441659614417, 8.554194504948999398194468424229, 9.544242905357488317009195169613, 10.32449259698026339247485307302, 11.91152298250087572989483998083