| L(s) = 1 | + 2.77·2-s + 0.269·3-s + 5.67·4-s + 0.747·6-s + 1.86·7-s + 10.1·8-s − 2.92·9-s − 3.25·11-s + 1.53·12-s − 3.40·13-s + 5.17·14-s + 16.8·16-s + 2.40·17-s − 8.10·18-s − 0.674·19-s + 0.504·21-s − 9.01·22-s − 7.41·23-s + 2.74·24-s − 9.42·26-s − 1.59·27-s + 10.6·28-s − 29-s + 5.25·31-s + 26.3·32-s − 0.877·33-s + 6.67·34-s + ⋯ |
| L(s) = 1 | + 1.95·2-s + 0.155·3-s + 2.83·4-s + 0.305·6-s + 0.706·7-s + 3.59·8-s − 0.975·9-s − 0.980·11-s + 0.442·12-s − 0.943·13-s + 1.38·14-s + 4.21·16-s + 0.584·17-s − 1.91·18-s − 0.154·19-s + 0.110·21-s − 1.92·22-s − 1.54·23-s + 0.560·24-s − 1.84·26-s − 0.307·27-s + 2.00·28-s − 0.185·29-s + 0.943·31-s + 4.65·32-s − 0.152·33-s + 1.14·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.013659498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.013659498\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 2.77T + 2T^{2} \) |
| 3 | \( 1 - 0.269T + 3T^{2} \) |
| 7 | \( 1 - 1.86T + 7T^{2} \) |
| 11 | \( 1 + 3.25T + 11T^{2} \) |
| 13 | \( 1 + 3.40T + 13T^{2} \) |
| 17 | \( 1 - 2.40T + 17T^{2} \) |
| 19 | \( 1 + 0.674T + 19T^{2} \) |
| 23 | \( 1 + 7.41T + 23T^{2} \) |
| 31 | \( 1 - 5.25T + 31T^{2} \) |
| 37 | \( 1 + 1.86T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 3.46T + 43T^{2} \) |
| 47 | \( 1 + 4.00T + 47T^{2} \) |
| 53 | \( 1 + 0.877T + 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 + 7.95T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 0.607T + 73T^{2} \) |
| 79 | \( 1 - 8.60T + 79T^{2} \) |
| 83 | \( 1 - 2.40T + 83T^{2} \) |
| 89 | \( 1 + 8.50T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−10.72723338715675530759290355344, −9.963555376483953160872413417673, −8.149568358986130733580655343282, −7.71154628272541474429051137057, −6.54226292755204087763584139394, −5.54934590252731235377886439729, −5.06518631589008421919996968373, −4.02400160088979750384325405108, −2.87549551626224488302870296613, −2.10599877537359630847455804246,
2.10599877537359630847455804246, 2.87549551626224488302870296613, 4.02400160088979750384325405108, 5.06518631589008421919996968373, 5.54934590252731235377886439729, 6.54226292755204087763584139394, 7.71154628272541474429051137057, 8.149568358986130733580655343282, 9.963555376483953160872413417673, 10.72723338715675530759290355344
