| L(s) = 1 | + 3-s + 1.56·5-s + 1.63·7-s + 9-s + 3.76·11-s − 1.49·13-s + 1.56·15-s − 7.60·17-s + 7.25·19-s + 1.63·21-s + 5.83·23-s − 2.53·25-s + 27-s + 7.42·29-s + 6.49·31-s + 3.76·33-s + 2.56·35-s − 4.26·37-s − 1.49·39-s + 5.80·41-s + 0.704·43-s + 1.56·45-s − 12.6·47-s − 4.33·49-s − 7.60·51-s + 1.19·53-s + 5.90·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.701·5-s + 0.617·7-s + 0.333·9-s + 1.13·11-s − 0.415·13-s + 0.405·15-s − 1.84·17-s + 1.66·19-s + 0.356·21-s + 1.21·23-s − 0.507·25-s + 0.192·27-s + 1.37·29-s + 1.16·31-s + 0.655·33-s + 0.433·35-s − 0.700·37-s − 0.240·39-s + 0.907·41-s + 0.107·43-s + 0.233·45-s − 1.84·47-s − 0.618·49-s − 1.06·51-s + 0.164·53-s + 0.796·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.539128317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.539128317\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 503 | \( 1 - T \) |
| good | 5 | \( 1 - 1.56T + 5T^{2} \) |
| 7 | \( 1 - 1.63T + 7T^{2} \) |
| 11 | \( 1 - 3.76T + 11T^{2} \) |
| 13 | \( 1 + 1.49T + 13T^{2} \) |
| 17 | \( 1 + 7.60T + 17T^{2} \) |
| 19 | \( 1 - 7.25T + 19T^{2} \) |
| 23 | \( 1 - 5.83T + 23T^{2} \) |
| 29 | \( 1 - 7.42T + 29T^{2} \) |
| 31 | \( 1 - 6.49T + 31T^{2} \) |
| 37 | \( 1 + 4.26T + 37T^{2} \) |
| 41 | \( 1 - 5.80T + 41T^{2} \) |
| 43 | \( 1 - 0.704T + 43T^{2} \) |
| 47 | \( 1 + 12.6T + 47T^{2} \) |
| 53 | \( 1 - 1.19T + 53T^{2} \) |
| 59 | \( 1 + 5.25T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 - 15.1T + 67T^{2} \) |
| 71 | \( 1 - 7.00T + 71T^{2} \) |
| 73 | \( 1 + 6.66T + 73T^{2} \) |
| 79 | \( 1 - 1.98T + 79T^{2} \) |
| 83 | \( 1 - 0.798T + 83T^{2} \) |
| 89 | \( 1 - 3.12T + 89T^{2} \) |
| 97 | \( 1 - 4.65T + 97T^{2} \) |
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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
−8.191127544397979111542334417959, −7.32877117267100923211552718120, −6.67045240790294794358082941544, −6.12648681817985113421765255851, −4.85619065384078537831405255075, −4.71372543089460483325047043707, −3.53729440915981920331147005298, −2.71709962236781410178580320706, −1.86487602118503773730372189094, −1.03613550756798650211312078414,
1.03613550756798650211312078414, 1.86487602118503773730372189094, 2.71709962236781410178580320706, 3.53729440915981920331147005298, 4.71372543089460483325047043707, 4.85619065384078537831405255075, 6.12648681817985113421765255851, 6.67045240790294794358082941544, 7.32877117267100923211552718120, 8.191127544397979111542334417959
