| L(s) = 1 | − 4.31·2-s + 10.6·4-s − 5·5-s − 11.3·8-s + 21.5·10-s − 19.7·11-s + 71.3·13-s − 36.0·16-s + 31.3·17-s + 136.·19-s − 53.1·20-s + 85.1·22-s + 100.·23-s + 25·25-s − 307.·26-s + 288.·29-s − 208.·31-s + 246.·32-s − 135.·34-s + 309.·37-s − 588.·38-s + 56.8·40-s + 181.·41-s − 18.2·43-s − 209.·44-s − 435.·46-s + 147.·47-s + ⋯ |
| L(s) = 1 | − 1.52·2-s + 1.32·4-s − 0.447·5-s − 0.502·8-s + 0.682·10-s − 0.540·11-s + 1.52·13-s − 0.562·16-s + 0.447·17-s + 1.64·19-s − 0.594·20-s + 0.825·22-s + 0.914·23-s + 0.200·25-s − 2.32·26-s + 1.84·29-s − 1.21·31-s + 1.36·32-s − 0.682·34-s + 1.37·37-s − 2.51·38-s + 0.224·40-s + 0.691·41-s − 0.0645·43-s − 0.718·44-s − 1.39·46-s + 0.458·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.164276101\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.164276101\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 4.31T + 8T^{2} \) |
| 11 | \( 1 + 19.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 71.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 31.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 136.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 100.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 288.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 208.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 309.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 181.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 18.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 147.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 127.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 322.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 341.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 84.3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 315.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.09e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.23e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 643.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.14e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.41e3T + 9.12e5T^{2} \) |
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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
−8.771467611427540362011245752386, −7.954346306536448158441352971162, −7.53661487272273927675004367515, −6.67589920278728255573896543101, −5.74637737247593893831252281694, −4.72478003916109560760391500051, −3.54681614247695077344597477653, −2.65715841504679139586836032888, −1.24268827967802971191827004604, −0.74089855284323588550253449091,
0.74089855284323588550253449091, 1.24268827967802971191827004604, 2.65715841504679139586836032888, 3.54681614247695077344597477653, 4.72478003916109560760391500051, 5.74637737247593893831252281694, 6.67589920278728255573896543101, 7.53661487272273927675004367515, 7.954346306536448158441352971162, 8.771467611427540362011245752386
