| L(s) = 1 | + 1.23·2-s − 0.468·4-s − 2.08·7-s − 3.05·8-s − 2.65·11-s + 0.149·13-s − 2.58·14-s − 2.84·16-s − 5.12·17-s − 3.08·19-s − 3.28·22-s + 2.76·23-s + 0.185·26-s + 0.977·28-s + 6.94·29-s + 1.52·31-s + 2.58·32-s − 6.34·34-s + 8.52·37-s − 3.81·38-s − 9.82·41-s + 3.43·43-s + 1.24·44-s + 3.41·46-s − 4.00·47-s − 2.64·49-s − 0.0701·52-s + ⋯ |
| L(s) = 1 | + 0.875·2-s − 0.234·4-s − 0.788·7-s − 1.08·8-s − 0.799·11-s + 0.0415·13-s − 0.690·14-s − 0.711·16-s − 1.24·17-s − 0.708·19-s − 0.699·22-s + 0.576·23-s + 0.0363·26-s + 0.184·28-s + 1.28·29-s + 0.273·31-s + 0.457·32-s − 1.08·34-s + 1.40·37-s − 0.619·38-s − 1.53·41-s + 0.523·43-s + 0.187·44-s + 0.504·46-s − 0.584·47-s − 0.378·49-s − 0.00972·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.437025782\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.437025782\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 1.23T + 2T^{2} \) |
| 7 | \( 1 + 2.08T + 7T^{2} \) |
| 11 | \( 1 + 2.65T + 11T^{2} \) |
| 13 | \( 1 - 0.149T + 13T^{2} \) |
| 17 | \( 1 + 5.12T + 17T^{2} \) |
| 19 | \( 1 + 3.08T + 19T^{2} \) |
| 23 | \( 1 - 2.76T + 23T^{2} \) |
| 29 | \( 1 - 6.94T + 29T^{2} \) |
| 31 | \( 1 - 1.52T + 31T^{2} \) |
| 37 | \( 1 - 8.52T + 37T^{2} \) |
| 41 | \( 1 + 9.82T + 41T^{2} \) |
| 43 | \( 1 - 3.43T + 43T^{2} \) |
| 47 | \( 1 + 4.00T + 47T^{2} \) |
| 53 | \( 1 + 0.945T + 53T^{2} \) |
| 59 | \( 1 - 8.18T + 59T^{2} \) |
| 61 | \( 1 + 4.47T + 61T^{2} \) |
| 67 | \( 1 + 3.34T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 - 3.45T + 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 + 0.0571T + 97T^{2} \) |
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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
−8.290975246142199362086244105338, −7.24754652029474011634466299977, −6.32480824461285745338867678814, −6.17588779855991099955942329303, −4.91427394034673833374276168506, −4.70262117822340865192183065153, −3.70414514425265817911464220894, −2.97149539164991883928776863529, −2.25680319654246704094234343203, −0.53026171254429675352790451848,
0.53026171254429675352790451848, 2.25680319654246704094234343203, 2.97149539164991883928776863529, 3.70414514425265817911464220894, 4.70262117822340865192183065153, 4.91427394034673833374276168506, 6.17588779855991099955942329303, 6.32480824461285745338867678814, 7.24754652029474011634466299977, 8.290975246142199362086244105338
