| L(s) = 1 | + 2-s + 0.149·3-s + 4-s + 1.63·5-s + 0.149·6-s + 8-s − 2.97·9-s + 1.63·10-s + 4.31·11-s + 0.149·12-s + 1.29·13-s + 0.244·15-s + 16-s + 6.73·17-s − 2.97·18-s + 4.24·19-s + 1.63·20-s + 4.31·22-s − 3.22·23-s + 0.149·24-s − 2.31·25-s + 1.29·26-s − 0.894·27-s + 3.58·29-s + 0.244·30-s − 10.4·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.0863·3-s + 0.5·4-s + 0.732·5-s + 0.0610·6-s + 0.353·8-s − 0.992·9-s + 0.517·10-s + 1.29·11-s + 0.0431·12-s + 0.358·13-s + 0.0632·15-s + 0.250·16-s + 1.63·17-s − 0.701·18-s + 0.972·19-s + 0.366·20-s + 0.919·22-s − 0.673·23-s + 0.0305·24-s − 0.463·25-s + 0.253·26-s − 0.172·27-s + 0.666·29-s + 0.0447·30-s − 1.87·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.482052238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.482052238\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 79 | \( 1 - T \) |
| good | 3 | \( 1 - 0.149T + 3T^{2} \) |
| 5 | \( 1 - 1.63T + 5T^{2} \) |
| 11 | \( 1 - 4.31T + 11T^{2} \) |
| 13 | \( 1 - 1.29T + 13T^{2} \) |
| 17 | \( 1 - 6.73T + 17T^{2} \) |
| 19 | \( 1 - 4.24T + 19T^{2} \) |
| 23 | \( 1 + 3.22T + 23T^{2} \) |
| 29 | \( 1 - 3.58T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 - 2.47T + 37T^{2} \) |
| 41 | \( 1 + 2.30T + 41T^{2} \) |
| 43 | \( 1 - 12.5T + 43T^{2} \) |
| 47 | \( 1 - 4.47T + 47T^{2} \) |
| 53 | \( 1 - 1.27T + 53T^{2} \) |
| 59 | \( 1 - 6.37T + 59T^{2} \) |
| 61 | \( 1 + 1.55T + 61T^{2} \) |
| 67 | \( 1 + 1.37T + 67T^{2} \) |
| 71 | \( 1 + 0.700T + 71T^{2} \) |
| 73 | \( 1 + 8.02T + 73T^{2} \) |
| 83 | \( 1 + 1.65T + 83T^{2} \) |
| 89 | \( 1 + 4.55T + 89T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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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
−7.70940362299056597882351326335, −7.12838507645477700036950875316, −6.13842803535556157922576690678, −5.75453394658233843510997486611, −5.34740407137227469524066861419, −4.14709284952544041697099858721, −3.56514079208300976614878395084, −2.84063317308015484796828904148, −1.86271189742630398587773131729, −1.00359531462023183536837710482,
1.00359531462023183536837710482, 1.86271189742630398587773131729, 2.84063317308015484796828904148, 3.56514079208300976614878395084, 4.14709284952544041697099858721, 5.34740407137227469524066861419, 5.75453394658233843510997486611, 6.13842803535556157922576690678, 7.12838507645477700036950875316, 7.70940362299056597882351326335
