| L(s) = 1 | − 2·2-s − 4.74·3-s + 4·4-s − 5·5-s + 9.49·6-s + 29.3·7-s − 8·8-s − 4.44·9-s + 10·10-s − 38.1·11-s − 18.9·12-s − 22.5·13-s − 58.7·14-s + 23.7·15-s + 16·16-s − 104.·17-s + 8.89·18-s + 141.·19-s − 20·20-s − 139.·21-s + 76.3·22-s − 23·23-s + 37.9·24-s + 25·25-s + 45.0·26-s + 149.·27-s + 117.·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.913·3-s + 0.5·4-s − 0.447·5-s + 0.646·6-s + 1.58·7-s − 0.353·8-s − 0.164·9-s + 0.316·10-s − 1.04·11-s − 0.456·12-s − 0.480·13-s − 1.12·14-s + 0.408·15-s + 0.250·16-s − 1.48·17-s + 0.116·18-s + 1.71·19-s − 0.223·20-s − 1.44·21-s + 0.739·22-s − 0.208·23-s + 0.323·24-s + 0.200·25-s + 0.340·26-s + 1.06·27-s + 0.792·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7948871915\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7948871915\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 + 5T \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 + 4.74T + 27T^{2} \) |
| 7 | \( 1 - 29.3T + 343T^{2} \) |
| 11 | \( 1 + 38.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 22.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 104.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 141.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 241.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 99.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 59.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 249.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 163.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 205.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 491.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 433.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 660.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 323.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 893.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 196.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 500.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 800.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 729.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.13e3T + 9.12e5T^{2} \) |
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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
−11.49432267756180009994680997296, −11.00805353501760541392368963471, −10.03958151604533505639337895230, −8.574655855616886075106987483725, −7.916229632306941277190238852518, −6.87539608483631960585577945622, −5.42243924147952364428299570897, −4.68229888209149143825694265876, −2.51179737146138258283283578057, −0.76139953237893540068576956796,
0.76139953237893540068576956796, 2.51179737146138258283283578057, 4.68229888209149143825694265876, 5.42243924147952364428299570897, 6.87539608483631960585577945622, 7.916229632306941277190238852518, 8.574655855616886075106987483725, 10.03958151604533505639337895230, 11.00805353501760541392368963471, 11.49432267756180009994680997296
