L(s) = 1 | + 3.17·3-s + 5-s − 0.749·7-s + 7.09·9-s + 5.92·11-s − 5.85·13-s + 3.17·15-s + 1.49·17-s + 3.53·19-s − 2.38·21-s + 0.831·23-s + 25-s + 13.0·27-s + 5.12·29-s + 8.73·31-s + 18.8·33-s − 0.749·35-s − 6.40·37-s − 18.6·39-s − 5.69·41-s + 6.93·43-s + 7.09·45-s − 2.32·47-s − 6.43·49-s + 4.76·51-s − 9.29·53-s + 5.92·55-s + ⋯ |
L(s) = 1 | + 1.83·3-s + 0.447·5-s − 0.283·7-s + 2.36·9-s + 1.78·11-s − 1.62·13-s + 0.820·15-s + 0.363·17-s + 0.811·19-s − 0.519·21-s + 0.173·23-s + 0.200·25-s + 2.50·27-s + 0.951·29-s + 1.56·31-s + 3.27·33-s − 0.126·35-s − 1.05·37-s − 2.97·39-s − 0.888·41-s + 1.05·43-s + 1.05·45-s − 0.339·47-s − 0.919·49-s + 0.667·51-s − 1.27·53-s + 0.798·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.982825711\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.982825711\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 - T \) |
good | 3 | \( 1 - 3.17T + 3T^{2} \) |
| 7 | \( 1 + 0.749T + 7T^{2} \) |
| 11 | \( 1 - 5.92T + 11T^{2} \) |
| 13 | \( 1 + 5.85T + 13T^{2} \) |
| 17 | \( 1 - 1.49T + 17T^{2} \) |
| 19 | \( 1 - 3.53T + 19T^{2} \) |
| 23 | \( 1 - 0.831T + 23T^{2} \) |
| 29 | \( 1 - 5.12T + 29T^{2} \) |
| 31 | \( 1 - 8.73T + 31T^{2} \) |
| 37 | \( 1 + 6.40T + 37T^{2} \) |
| 41 | \( 1 + 5.69T + 41T^{2} \) |
| 43 | \( 1 - 6.93T + 43T^{2} \) |
| 47 | \( 1 + 2.32T + 47T^{2} \) |
| 53 | \( 1 + 9.29T + 53T^{2} \) |
| 59 | \( 1 - 3.99T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 + 6.62T + 67T^{2} \) |
| 71 | \( 1 - 3.69T + 71T^{2} \) |
| 73 | \( 1 - 1.95T + 73T^{2} \) |
| 79 | \( 1 - 2.81T + 79T^{2} \) |
| 83 | \( 1 + 17.5T + 83T^{2} \) |
| 89 | \( 1 + 6.27T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−8.167631029198573615278976332254, −7.40093744349002691279955792953, −6.88536360405237823952775505759, −6.19950398655799805319114481220, −4.87380771512045236080038238066, −4.36490587423552886035306655721, −3.30497511880225008068112298975, −2.95285464620906213175894026413, −1.96699692709095836047221299563, −1.18875532363804678772836889388,
1.18875532363804678772836889388, 1.96699692709095836047221299563, 2.95285464620906213175894026413, 3.30497511880225008068112298975, 4.36490587423552886035306655721, 4.87380771512045236080038238066, 6.19950398655799805319114481220, 6.88536360405237823952775505759, 7.40093744349002691279955792953, 8.167631029198573615278976332254