| L(s) = 1 | + 0.529·2-s + 3·3-s − 7.71·4-s + 5·5-s + 1.58·6-s + 30.0·7-s − 8.32·8-s + 9·9-s + 2.64·10-s + 11.5·11-s − 23.1·12-s − 64.3·13-s + 15.9·14-s + 15·15-s + 57.3·16-s + 29.8·17-s + 4.76·18-s + 103.·19-s − 38.5·20-s + 90.2·21-s + 6.12·22-s − 29.0·23-s − 24.9·24-s + 25·25-s − 34.0·26-s + 27·27-s − 232.·28-s + ⋯ |
| L(s) = 1 | + 0.187·2-s + 0.577·3-s − 0.964·4-s + 0.447·5-s + 0.108·6-s + 1.62·7-s − 0.367·8-s + 0.333·9-s + 0.0837·10-s + 0.317·11-s − 0.557·12-s − 1.37·13-s + 0.304·14-s + 0.258·15-s + 0.896·16-s + 0.426·17-s + 0.0624·18-s + 1.24·19-s − 0.431·20-s + 0.937·21-s + 0.0593·22-s − 0.263·23-s − 0.212·24-s + 0.200·25-s − 0.257·26-s + 0.192·27-s − 1.56·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.724223060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.724223060\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 - 5T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 - 0.529T + 8T^{2} \) |
| 7 | \( 1 - 30.0T + 343T^{2} \) |
| 11 | \( 1 - 11.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 64.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 29.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 103.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 29.0T + 1.21e4T^{2} \) |
| 31 | \( 1 + 44.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 21.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 35.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 367.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 504.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 290.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 168.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 287.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 388.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 337.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 487.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 625.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 747.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 205.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.08e3T + 9.12e5T^{2} \) |
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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
−10.57332805094497608685072651851, −9.618239565442778688380682482095, −9.009451725558400980641770977491, −7.963764304028980761829979869051, −7.37293294718435019986943380447, −5.57565102745898495928118912989, −4.92462915062139852645189310350, −3.95257316214826927749233417066, −2.45498459206126874728474010141, −1.10141587077213584111898626252,
1.10141587077213584111898626252, 2.45498459206126874728474010141, 3.95257316214826927749233417066, 4.92462915062139852645189310350, 5.57565102745898495928118912989, 7.37293294718435019986943380447, 7.963764304028980761829979869051, 9.009451725558400980641770977491, 9.618239565442778688380682482095, 10.57332805094497608685072651851
