| L(s) = 1 | + 5.02·2-s + 17.2·4-s + 11.3·5-s + 16.2·7-s + 46.4·8-s + 57.2·10-s + 28.0·11-s + 68.1·13-s + 81.6·14-s + 95.3·16-s − 69.3·17-s + 23.2·19-s + 196.·20-s + 140.·22-s + 109.·23-s + 4.94·25-s + 342.·26-s + 280.·28-s − 90.9·29-s − 27.9·31-s + 107.·32-s − 348.·34-s + 185.·35-s + 146.·37-s + 116.·38-s + 529.·40-s − 353.·41-s + ⋯ |
| L(s) = 1 | + 1.77·2-s + 2.15·4-s + 1.01·5-s + 0.877·7-s + 2.05·8-s + 1.81·10-s + 0.769·11-s + 1.45·13-s + 1.55·14-s + 1.48·16-s − 0.989·17-s + 0.280·19-s + 2.19·20-s + 1.36·22-s + 0.995·23-s + 0.0395·25-s + 2.58·26-s + 1.89·28-s − 0.582·29-s − 0.162·31-s + 0.594·32-s − 1.75·34-s + 0.894·35-s + 0.648·37-s + 0.498·38-s + 2.09·40-s − 1.34·41-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
| \(L(2)\) |
\(\approx\) |
\(11.16946229\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.16946229\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 5.02T + 8T^{2} \) |
| 5 | \( 1 - 11.3T + 125T^{2} \) |
| 7 | \( 1 - 16.2T + 343T^{2} \) |
| 11 | \( 1 - 28.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 68.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 69.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 23.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 109.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 90.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 27.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 146.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 353.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 365.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 283.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 666.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 33.7T + 2.05e5T^{2} \) |
| 61 | \( 1 - 30.7T + 2.26e5T^{2} \) |
| 67 | \( 1 + 75.6T + 3.00e5T^{2} \) |
| 71 | \( 1 + 872.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 907.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 252.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 289.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 866.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 256.T + 9.12e5T^{2} \) |
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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
−8.758084838478621754112316881494, −7.69870113226771442741714916295, −6.61658240792771415880933725153, −6.25328176169206363651937954188, −5.44617611348952666209793697441, −4.74808137218153400903416638195, −3.95673254116904789275521069382, −3.09223195649326670876871501556, −1.96396987271756445868653576604, −1.37281943408789008389661205334,
1.37281943408789008389661205334, 1.96396987271756445868653576604, 3.09223195649326670876871501556, 3.95673254116904789275521069382, 4.74808137218153400903416638195, 5.44617611348952666209793697441, 6.25328176169206363651937954188, 6.61658240792771415880933725153, 7.69870113226771442741714916295, 8.758084838478621754112316881494
