| L(s) = 1 | + 4.72·2-s − 9.01·3-s + 14.3·4-s − 1.14·5-s − 42.5·6-s + 29.8·8-s + 54.2·9-s − 5.40·10-s + 46.3·11-s − 129.·12-s + 78.5·13-s + 10.3·15-s + 26.5·16-s + 117.·17-s + 256.·18-s + 34.0·19-s − 16.3·20-s + 218.·22-s − 58.1·23-s − 269.·24-s − 123.·25-s + 371.·26-s − 245.·27-s + 219.·29-s + 48.7·30-s − 312.·31-s − 113.·32-s + ⋯ |
| L(s) = 1 | + 1.67·2-s − 1.73·3-s + 1.79·4-s − 0.102·5-s − 2.89·6-s + 1.31·8-s + 2.00·9-s − 0.170·10-s + 1.27·11-s − 3.10·12-s + 1.67·13-s + 0.177·15-s + 0.414·16-s + 1.67·17-s + 3.35·18-s + 0.411·19-s − 0.183·20-s + 2.12·22-s − 0.527·23-s − 2.28·24-s − 0.989·25-s + 2.79·26-s − 1.75·27-s + 1.40·29-s + 0.296·30-s − 1.81·31-s − 0.627·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.408758090\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.408758090\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 47 | \( 1 - 47T \) |
| good | 2 | \( 1 - 4.72T + 8T^{2} \) |
| 3 | \( 1 + 9.01T + 27T^{2} \) |
| 5 | \( 1 + 1.14T + 125T^{2} \) |
| 11 | \( 1 - 46.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 78.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 117.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 34.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 58.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 219.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 312.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 189.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 193.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 374.T + 7.95e4T^{2} \) |
| 53 | \( 1 - 680.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 120.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 750.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 132.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 536.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 766.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 809.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 154.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 761.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.17e3T + 9.12e5T^{2} \) |
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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
−8.535700990567859261285902397240, −7.31483869997125945327940443880, −6.64184803716089732219938794902, −5.87580401758247079358571580421, −5.72071399832034240374743221585, −4.78107369217433214254738578082, −3.84208847922592140607221042755, −3.51793794066497543299318066814, −1.68708097087718232712015755041, −0.861222237274731647610419394318,
0.861222237274731647610419394318, 1.68708097087718232712015755041, 3.51793794066497543299318066814, 3.84208847922592140607221042755, 4.78107369217433214254738578082, 5.72071399832034240374743221585, 5.87580401758247079358571580421, 6.64184803716089732219938794902, 7.31483869997125945327940443880, 8.535700990567859261285902397240
