| L(s) = 1 | − 5.26·2-s + 19.6·4-s + 10.3·7-s − 61.4·8-s − 11·11-s − 63.9·13-s − 54.3·14-s + 165.·16-s + 17.1·17-s + 90.2·19-s + 57.8·22-s − 212.·23-s + 336.·26-s + 203.·28-s − 57.5·29-s − 141.·31-s − 381.·32-s − 90.2·34-s + 257.·37-s − 474.·38-s + 225.·41-s + 347.·43-s − 216.·44-s + 1.11e3·46-s + 404.·47-s − 236.·49-s − 1.25e3·52-s + ⋯ |
| L(s) = 1 | − 1.86·2-s + 2.46·4-s + 0.557·7-s − 2.71·8-s − 0.301·11-s − 1.36·13-s − 1.03·14-s + 2.59·16-s + 0.244·17-s + 1.08·19-s + 0.560·22-s − 1.92·23-s + 2.53·26-s + 1.37·28-s − 0.368·29-s − 0.820·31-s − 2.10·32-s − 0.455·34-s + 1.14·37-s − 2.02·38-s + 0.860·41-s + 1.23·43-s − 0.741·44-s + 3.58·46-s + 1.25·47-s − 0.689·49-s − 3.35·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 5.26T + 8T^{2} \) |
| 7 | \( 1 - 10.3T + 343T^{2} \) |
| 13 | \( 1 + 63.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 17.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 90.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 212.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 57.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 141.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 257.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 225.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 347.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 404.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 259.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 853.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 203.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 266.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 92.4T + 3.57e5T^{2} \) |
| 73 | \( 1 - 242.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.02e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 706.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 440.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 197.T + 9.12e5T^{2} \) |
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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
−8.091805856604266478594351273548, −7.62092462870230603397638179250, −7.22355685569155139625425305056, −6.05909483957842277427810226753, −5.35079208258557067545549464792, −4.05722170900786590029058753528, −2.66639291894470737845918107640, −2.07528586890735883790936705105, −0.981325976330552932039369362914, 0,
0.981325976330552932039369362914, 2.07528586890735883790936705105, 2.66639291894470737845918107640, 4.05722170900786590029058753528, 5.35079208258557067545549464792, 6.05909483957842277427810226753, 7.22355685569155139625425305056, 7.62092462870230603397638179250, 8.091805856604266478594351273548
