| L(s) = 1 | + 2.30·2-s + 3.30·4-s − 3·5-s + 3.00·8-s − 6.90·10-s + 0.697·11-s + 5.60·13-s + 0.302·16-s − 5.30·17-s − 19-s − 9.90·20-s + 1.60·22-s + 3·23-s + 4·25-s + 12.9·26-s − 9.90·29-s − 1.30·31-s − 5.30·32-s − 12.2·34-s + 3.60·37-s − 2.30·38-s − 9.00·40-s + 0.697·41-s − 10·43-s + 2.30·44-s + 6.90·46-s + 6.21·47-s + ⋯ |
| L(s) = 1 | + 1.62·2-s + 1.65·4-s − 1.34·5-s + 1.06·8-s − 2.18·10-s + 0.210·11-s + 1.55·13-s + 0.0756·16-s − 1.28·17-s − 0.229·19-s − 2.21·20-s + 0.342·22-s + 0.625·23-s + 0.800·25-s + 2.53·26-s − 1.83·29-s − 0.233·31-s − 0.937·32-s − 2.09·34-s + 0.592·37-s − 0.373·38-s − 1.42·40-s + 0.108·41-s − 1.52·43-s + 0.347·44-s + 1.01·46-s + 0.905·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 2.30T + 2T^{2} \) |
| 5 | \( 1 + 3T + 5T^{2} \) |
| 11 | \( 1 - 0.697T + 11T^{2} \) |
| 13 | \( 1 - 5.60T + 13T^{2} \) |
| 17 | \( 1 + 5.30T + 17T^{2} \) |
| 23 | \( 1 - 3T + 23T^{2} \) |
| 29 | \( 1 + 9.90T + 29T^{2} \) |
| 31 | \( 1 + 1.30T + 31T^{2} \) |
| 37 | \( 1 - 3.60T + 37T^{2} \) |
| 41 | \( 1 - 0.697T + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 - 6.21T + 47T^{2} \) |
| 53 | \( 1 - 6.90T + 53T^{2} \) |
| 59 | \( 1 + 6.21T + 59T^{2} \) |
| 61 | \( 1 - 4.21T + 61T^{2} \) |
| 67 | \( 1 + 1.90T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 + 1.51T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 + 9.60T + 97T^{2} \) |
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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
−7.18604856433000859325774601653, −6.65011762425711227739276756987, −5.95983835796328862605639317692, −5.26643689857157167913668966383, −4.36878364394392104001172225143, −3.95845000724784734236020876983, −3.49795014589698163845837669879, −2.61212895227803993785347583972, −1.52000677817342647957944882485, 0,
1.52000677817342647957944882485, 2.61212895227803993785347583972, 3.49795014589698163845837669879, 3.95845000724784734236020876983, 4.36878364394392104001172225143, 5.26643689857157167913668966383, 5.95983835796328862605639317692, 6.65011762425711227739276756987, 7.18604856433000859325774601653
