| L(s) = 1 | + 2.64·2-s + 3-s + 5.00·4-s + 5-s + 2.64·6-s + 1.64·7-s + 7.93·8-s + 9-s + 2.64·10-s + 0.354·11-s + 5.00·12-s + 0.354·13-s + 4.35·14-s + 15-s + 11.0·16-s − 4·17-s + 2.64·18-s + 5.00·20-s + 1.64·21-s + 0.937·22-s + 9.29·23-s + 7.93·24-s + 25-s + 0.937·26-s + 27-s + 8.22·28-s − 8.93·29-s + ⋯ |
| L(s) = 1 | + 1.87·2-s + 0.577·3-s + 2.50·4-s + 0.447·5-s + 1.08·6-s + 0.622·7-s + 2.80·8-s + 0.333·9-s + 0.836·10-s + 0.106·11-s + 1.44·12-s + 0.0982·13-s + 1.16·14-s + 0.258·15-s + 2.75·16-s − 0.970·17-s + 0.623·18-s + 1.11·20-s + 0.359·21-s + 0.199·22-s + 1.93·23-s + 1.62·24-s + 0.200·25-s + 0.183·26-s + 0.192·27-s + 1.55·28-s − 1.65·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.04130909\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.04130909\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 2.64T + 2T^{2} \) |
| 7 | \( 1 - 1.64T + 7T^{2} \) |
| 11 | \( 1 - 0.354T + 11T^{2} \) |
| 13 | \( 1 - 0.354T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 23 | \( 1 - 9.29T + 23T^{2} \) |
| 29 | \( 1 + 8.93T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 3.64T + 37T^{2} \) |
| 41 | \( 1 - 9.64T + 41T^{2} \) |
| 43 | \( 1 - 5.64T + 43T^{2} \) |
| 47 | \( 1 + 1.29T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 6.58T + 67T^{2} \) |
| 71 | \( 1 + 7.29T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 6.58T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 - 16.9T + 89T^{2} \) |
| 97 | \( 1 - 2.93T + 97T^{2} \) |
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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
−7.73628304132250010195559367429, −7.34446391802191419039812226846, −6.51981573498064807130381809168, −5.88487183819479583497450111514, −5.02467014561362178787827386304, −4.62309372226598258051105854047, −3.71670326153197024388569019651, −3.04594607498498557803039185198, −2.17789635837277201870717159396, −1.51476721695981464304598333712,
1.51476721695981464304598333712, 2.17789635837277201870717159396, 3.04594607498498557803039185198, 3.71670326153197024388569019651, 4.62309372226598258051105854047, 5.02467014561362178787827386304, 5.88487183819479583497450111514, 6.51981573498064807130381809168, 7.34446391802191419039812226846, 7.73628304132250010195559367429
