| L(s) = 1 | + 1.32·3-s + 2.50·5-s + 4.14·7-s − 1.24·9-s + 4.74·11-s + 1.40·13-s + 3.32·15-s + 2.38·17-s − 5.99·19-s + 5.49·21-s − 0.793·23-s + 1.28·25-s − 5.62·27-s − 6.43·29-s − 5.18·31-s + 6.28·33-s + 10.3·35-s + 0.565·37-s + 1.86·39-s + 8.01·41-s + 10.1·43-s − 3.11·45-s + 1.44·47-s + 10.1·49-s + 3.15·51-s + 6.79·53-s + 11.8·55-s + ⋯ |
| L(s) = 1 | + 0.765·3-s + 1.12·5-s + 1.56·7-s − 0.413·9-s + 1.42·11-s + 0.390·13-s + 0.858·15-s + 0.577·17-s − 1.37·19-s + 1.19·21-s − 0.165·23-s + 0.256·25-s − 1.08·27-s − 1.19·29-s − 0.931·31-s + 1.09·33-s + 1.75·35-s + 0.0929·37-s + 0.299·39-s + 1.25·41-s + 1.54·43-s − 0.463·45-s + 0.210·47-s + 1.45·49-s + 0.442·51-s + 0.933·53-s + 1.60·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.557141862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.557141862\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 - 1.32T + 3T^{2} \) |
| 5 | \( 1 - 2.50T + 5T^{2} \) |
| 7 | \( 1 - 4.14T + 7T^{2} \) |
| 11 | \( 1 - 4.74T + 11T^{2} \) |
| 13 | \( 1 - 1.40T + 13T^{2} \) |
| 17 | \( 1 - 2.38T + 17T^{2} \) |
| 19 | \( 1 + 5.99T + 19T^{2} \) |
| 23 | \( 1 + 0.793T + 23T^{2} \) |
| 29 | \( 1 + 6.43T + 29T^{2} \) |
| 31 | \( 1 + 5.18T + 31T^{2} \) |
| 37 | \( 1 - 0.565T + 37T^{2} \) |
| 41 | \( 1 - 8.01T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 - 1.44T + 47T^{2} \) |
| 53 | \( 1 - 6.79T + 53T^{2} \) |
| 59 | \( 1 + 7.29T + 59T^{2} \) |
| 61 | \( 1 - 5.02T + 61T^{2} \) |
| 67 | \( 1 + 2.39T + 67T^{2} \) |
| 71 | \( 1 - 2.82T + 71T^{2} \) |
| 73 | \( 1 - 9.12T + 73T^{2} \) |
| 79 | \( 1 + 1.33T + 79T^{2} \) |
| 83 | \( 1 - 16.7T + 83T^{2} \) |
| 89 | \( 1 + 5.45T + 89T^{2} \) |
| 97 | \( 1 - 18.3T + 97T^{2} \) |
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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
−7.85981881489396293345383754709, −7.36482393053646959747023712098, −6.20235240876916707313490198394, −5.89011873427861524994692549338, −5.08989390062194405672051354992, −4.12306335452025965393673365607, −3.64613595464139852919337615774, −2.27047139246088021706856640326, −2.00097073712572679870342546674, −1.10295018887541718736149314185,
1.10295018887541718736149314185, 2.00097073712572679870342546674, 2.27047139246088021706856640326, 3.64613595464139852919337615774, 4.12306335452025965393673365607, 5.08989390062194405672051354992, 5.89011873427861524994692549338, 6.20235240876916707313490198394, 7.36482393053646959747023712098, 7.85981881489396293345383754709
