| L(s) = 1 | + 2·2-s + 6.93·3-s + 4·4-s + 13.8·6-s − 7.91·7-s + 8·8-s + 21.1·9-s + 26.0·11-s + 27.7·12-s + 31.1·13-s − 15.8·14-s + 16·16-s + 24.1·17-s + 42.2·18-s + 85.8·19-s − 54.9·21-s + 52.1·22-s − 23·23-s + 55.5·24-s + 62.2·26-s − 40.7·27-s − 31.6·28-s + 195.·29-s − 9.44·31-s + 32·32-s + 180.·33-s + 48.3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.33·3-s + 0.5·4-s + 0.944·6-s − 0.427·7-s + 0.353·8-s + 0.782·9-s + 0.714·11-s + 0.667·12-s + 0.663·13-s − 0.302·14-s + 0.250·16-s + 0.344·17-s + 0.553·18-s + 1.03·19-s − 0.570·21-s + 0.505·22-s − 0.208·23-s + 0.472·24-s + 0.469·26-s − 0.290·27-s − 0.213·28-s + 1.25·29-s − 0.0546·31-s + 0.176·32-s + 0.953·33-s + 0.243·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.001264875\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.001264875\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 - 6.93T + 27T^{2} \) |
| 7 | \( 1 + 7.91T + 343T^{2} \) |
| 11 | \( 1 - 26.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 31.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 24.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 85.8T + 6.85e3T^{2} \) |
| 29 | \( 1 - 195.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 9.44T + 2.97e4T^{2} \) |
| 37 | \( 1 + 126.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 102.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 112.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 162.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 88.3T + 1.48e5T^{2} \) |
| 59 | \( 1 + 397.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 414.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 570.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 212.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 942.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 673.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 194.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 273.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 123.T + 9.12e5T^{2} \) |
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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
−9.377766327291878626100445534475, −8.562755675541791477711887399588, −7.83941238719816327605730560596, −6.91338367915435321939891212337, −6.11258060265963177702732577199, −5.01473932954440124685517409381, −3.79781520010163009295261432747, −3.35166467791110580459723137647, −2.37032161220025771425283393236, −1.16488085269865273444683663173,
1.16488085269865273444683663173, 2.37032161220025771425283393236, 3.35166467791110580459723137647, 3.79781520010163009295261432747, 5.01473932954440124685517409381, 6.11258060265963177702732577199, 6.91338367915435321939891212337, 7.83941238719816327605730560596, 8.562755675541791477711887399588, 9.377766327291878626100445534475
