L(s) = 1 | − 2.41·2-s + 0.757·3-s − 2.17·4-s + 10.6·5-s − 1.82·6-s + 24.5·8-s − 26.4·9-s − 25.7·10-s + 39.3·11-s − 1.64·12-s − 23.7·13-s + 8.07·15-s − 41.9·16-s − 4.54·17-s + 63.7·18-s + 155.·19-s − 23.1·20-s − 94.9·22-s − 41.8·23-s + 18.5·24-s − 11.4·25-s + 57.3·26-s − 40.4·27-s + 29·29-s − 19.4·30-s + 57.9·31-s − 95.2·32-s + ⋯ |
L(s) = 1 | − 0.853·2-s + 0.145·3-s − 0.271·4-s + 0.953·5-s − 0.124·6-s + 1.08·8-s − 0.978·9-s − 0.813·10-s + 1.07·11-s − 0.0395·12-s − 0.507·13-s + 0.138·15-s − 0.654·16-s − 0.0648·17-s + 0.835·18-s + 1.87·19-s − 0.258·20-s − 0.920·22-s − 0.379·23-s + 0.158·24-s − 0.0914·25-s + 0.432·26-s − 0.288·27-s + 0.185·29-s − 0.118·30-s + 0.335·31-s − 0.526·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1421 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1421 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.468605835\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.468605835\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 29 | \( 1 - 29T \) |
good | 2 | \( 1 + 2.41T + 8T^{2} \) |
| 3 | \( 1 - 0.757T + 27T^{2} \) |
| 5 | \( 1 - 10.6T + 125T^{2} \) |
| 11 | \( 1 - 39.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 23.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 4.54T + 4.91e3T^{2} \) |
| 19 | \( 1 - 155.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 41.8T + 1.21e4T^{2} \) |
| 31 | \( 1 - 57.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 235.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 175.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 402.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 227.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 673.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 800.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 222.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 524.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 281.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.22e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 611.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 515.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 358.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 829.T + 9.12e5T^{2} \) |
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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
−9.276711027024936482688989358161, −8.576319601095259001868918791479, −7.76378137545148771228603598276, −6.88336093772848305249775681240, −5.84828577779909208480906640299, −5.16973407631350342054151336092, −4.02832143714565877389367044259, −2.84264723099625218206690757567, −1.69913191975351775496309056745, −0.70966006069231470176172940773,
0.70966006069231470176172940773, 1.69913191975351775496309056745, 2.84264723099625218206690757567, 4.02832143714565877389367044259, 5.16973407631350342054151336092, 5.84828577779909208480906640299, 6.88336093772848305249775681240, 7.76378137545148771228603598276, 8.576319601095259001868918791479, 9.276711027024936482688989358161