L(s) = 1 | − 2.96·2-s + 0.781·4-s + 9.17·5-s + 2.35·7-s + 21.3·8-s − 27.1·10-s − 14.3·11-s + 15.1·13-s − 6.98·14-s − 69.6·16-s − 57.8·17-s + 103.·19-s + 7.16·20-s + 42.6·22-s + 82.7·23-s − 40.8·25-s − 44.8·26-s + 1.84·28-s − 82.2·29-s − 156.·31-s + 35.2·32-s + 171.·34-s + 21.6·35-s + 202.·37-s − 305.·38-s + 196.·40-s − 3.54·41-s + ⋯ |
L(s) = 1 | − 1.04·2-s + 0.0976·4-s + 0.820·5-s + 0.127·7-s + 0.945·8-s − 0.859·10-s − 0.394·11-s + 0.323·13-s − 0.133·14-s − 1.08·16-s − 0.825·17-s + 1.24·19-s + 0.0801·20-s + 0.412·22-s + 0.749·23-s − 0.326·25-s − 0.338·26-s + 0.0124·28-s − 0.526·29-s − 0.903·31-s + 0.194·32-s + 0.865·34-s + 0.104·35-s + 0.897·37-s − 1.30·38-s + 0.775·40-s − 0.0135·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.273020215\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.273020215\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 239 | \( 1 + 239T \) |
good | 2 | \( 1 + 2.96T + 8T^{2} \) |
| 5 | \( 1 - 9.17T + 125T^{2} \) |
| 7 | \( 1 - 2.35T + 343T^{2} \) |
| 11 | \( 1 + 14.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 15.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 57.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 103.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 82.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 82.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 156.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 202.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 3.54T + 6.89e4T^{2} \) |
| 43 | \( 1 - 357.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 147.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 160.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 45.9T + 2.05e5T^{2} \) |
| 61 | \( 1 + 743.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 165.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 747.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 962.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 803.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 588.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 764.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 135.T + 9.12e5T^{2} \) |
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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
−9.008748201845867032131268069557, −7.968753209781256782249377078197, −7.46132038504238157839997186157, −6.52196694984941439604080141020, −5.56350411533969692921015891173, −4.86301506467547040243194162534, −3.78017887354710519753273265585, −2.49721228962470240705072543746, −1.59899075157608011056453003878, −0.62859666203649286475057852295,
0.62859666203649286475057852295, 1.59899075157608011056453003878, 2.49721228962470240705072543746, 3.78017887354710519753273265585, 4.86301506467547040243194162534, 5.56350411533969692921015891173, 6.52196694984941439604080141020, 7.46132038504238157839997186157, 7.968753209781256782249377078197, 9.008748201845867032131268069557