| L(s) = 1 | + 2·2-s − 0.377·3-s + 4·4-s + 19.5·5-s − 0.755·6-s + 8·8-s − 26.8·9-s + 39.1·10-s + 30.9·11-s − 1.51·12-s − 7.34·13-s − 7.39·15-s + 16·16-s − 20.0·17-s − 53.7·18-s − 143.·19-s + 78.3·20-s + 61.9·22-s − 23·23-s − 3.02·24-s + 258.·25-s − 14.6·26-s + 20.3·27-s − 205.·29-s − 14.7·30-s + 265.·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.0726·3-s + 0.5·4-s + 1.75·5-s − 0.0514·6-s + 0.353·8-s − 0.994·9-s + 1.23·10-s + 0.848·11-s − 0.0363·12-s − 0.156·13-s − 0.127·15-s + 0.250·16-s − 0.286·17-s − 0.703·18-s − 1.73·19-s + 0.875·20-s + 0.599·22-s − 0.208·23-s − 0.0257·24-s + 2.06·25-s − 0.110·26-s + 0.145·27-s − 1.31·29-s − 0.0900·30-s + 1.53·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.027162441\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.027162441\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 + 0.377T + 27T^{2} \) |
| 5 | \( 1 - 19.5T + 125T^{2} \) |
| 11 | \( 1 - 30.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 7.34T + 2.19e3T^{2} \) |
| 17 | \( 1 + 20.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 143.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 205.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 265.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 327.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 315.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 275.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 428.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 744.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 466.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 86.8T + 2.26e5T^{2} \) |
| 67 | \( 1 + 104.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 621.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 282.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 147.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.02e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.30e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 688.T + 9.12e5T^{2} \) |
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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
−8.989546863204920698047909432959, −7.86387699259925951215615808203, −6.69309584413035364297088472894, −6.06720177405040410695779153424, −5.81034144608028675382151889111, −4.74177587563572110042045655387, −3.90717859352312963950814228211, −2.48797384759666895081631193326, −2.24686786223198547529609610175, −0.912592748013112673546226205764,
0.912592748013112673546226205764, 2.24686786223198547529609610175, 2.48797384759666895081631193326, 3.90717859352312963950814228211, 4.74177587563572110042045655387, 5.81034144608028675382151889111, 6.06720177405040410695779153424, 6.69309584413035364297088472894, 7.86387699259925951215615808203, 8.989546863204920698047909432959
