| L(s) = 1 | − 3.35·2-s − 3·3-s + 3.28·4-s + 10.0·6-s − 30.4·7-s + 15.8·8-s + 9·9-s + 31.4·11-s − 9.84·12-s + 60.7·13-s + 102.·14-s − 79.4·16-s + 121.·17-s − 30.2·18-s − 14.4·19-s + 91.3·21-s − 105.·22-s − 13.6·23-s − 47.5·24-s − 204.·26-s − 27·27-s − 99.8·28-s − 76.0·29-s + 183.·31-s + 140.·32-s − 94.3·33-s − 407.·34-s + ⋯ |
| L(s) = 1 | − 1.18·2-s − 0.577·3-s + 0.410·4-s + 0.685·6-s − 1.64·7-s + 0.700·8-s + 0.333·9-s + 0.861·11-s − 0.236·12-s + 1.29·13-s + 1.95·14-s − 1.24·16-s + 1.72·17-s − 0.395·18-s − 0.174·19-s + 0.948·21-s − 1.02·22-s − 0.124·23-s − 0.404·24-s − 1.53·26-s − 0.192·27-s − 0.674·28-s − 0.486·29-s + 1.06·31-s + 0.774·32-s − 0.497·33-s − 2.05·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5601919483\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5601919483\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 3.35T + 8T^{2} \) |
| 7 | \( 1 + 30.4T + 343T^{2} \) |
| 11 | \( 1 - 31.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 60.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 121.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 14.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 13.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 76.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 183.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 37.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 30.6T + 6.89e4T^{2} \) |
| 43 | \( 1 + 327.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 449.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 301.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 340.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 619.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 256.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 499.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 19.1T + 3.89e5T^{2} \) |
| 79 | \( 1 - 257.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 914.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.05e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 521T + 9.12e5T^{2} \) |
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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
−13.88639904392825721058657660870, −12.84155207522660950040645693125, −11.64513536298935963850060994379, −10.29743689672137619265614504293, −9.658630696943240842892968042720, −8.527440066004152581283656322330, −7.01964205401379894543070445435, −5.94500376126656922686914867451, −3.71957028389663821607026622703, −0.911195730578247287836466304871,
0.911195730578247287836466304871, 3.71957028389663821607026622703, 5.94500376126656922686914867451, 7.01964205401379894543070445435, 8.527440066004152581283656322330, 9.658630696943240842892968042720, 10.29743689672137619265614504293, 11.64513536298935963850060994379, 12.84155207522660950040645693125, 13.88639904392825721058657660870
