| L(s) = 1 | − 1.53·2-s − 3-s + 0.369·4-s + 1.53·6-s − 4.87·7-s + 2.51·8-s + 9-s + 4.34·11-s − 0.369·12-s + 2.53·13-s + 7.51·14-s − 4.60·16-s − 2.63·17-s − 1.53·18-s − 7.41·19-s + 4.87·21-s − 6.68·22-s − 2.29·23-s − 2.51·24-s − 3.90·26-s − 27-s − 1.80·28-s + 6.09·29-s − 31-s + 2.06·32-s − 4.34·33-s + 4.04·34-s + ⋯ |
| L(s) = 1 | − 1.08·2-s − 0.577·3-s + 0.184·4-s + 0.628·6-s − 1.84·7-s + 0.887·8-s + 0.333·9-s + 1.30·11-s − 0.106·12-s + 0.704·13-s + 2.00·14-s − 1.15·16-s − 0.638·17-s − 0.362·18-s − 1.70·19-s + 1.06·21-s − 1.42·22-s − 0.477·23-s − 0.512·24-s − 0.766·26-s − 0.192·27-s − 0.340·28-s + 1.13·29-s − 0.179·31-s + 0.364·32-s − 0.755·33-s + 0.694·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 1.53T + 2T^{2} \) |
| 7 | \( 1 + 4.87T + 7T^{2} \) |
| 11 | \( 1 - 4.34T + 11T^{2} \) |
| 13 | \( 1 - 2.53T + 13T^{2} \) |
| 17 | \( 1 + 2.63T + 17T^{2} \) |
| 19 | \( 1 + 7.41T + 19T^{2} \) |
| 23 | \( 1 + 2.29T + 23T^{2} \) |
| 29 | \( 1 - 6.09T + 29T^{2} \) |
| 37 | \( 1 - 5.80T + 37T^{2} \) |
| 41 | \( 1 + 0.183T + 41T^{2} \) |
| 43 | \( 1 - 6.49T + 43T^{2} \) |
| 47 | \( 1 - 9.80T + 47T^{2} \) |
| 53 | \( 1 - 1.86T + 53T^{2} \) |
| 59 | \( 1 + 7.90T + 59T^{2} \) |
| 61 | \( 1 + 8.15T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 - 3.17T + 71T^{2} \) |
| 73 | \( 1 - 15.5T + 73T^{2} \) |
| 79 | \( 1 + 6.23T + 79T^{2} \) |
| 83 | \( 1 + 6.38T + 83T^{2} \) |
| 89 | \( 1 + 7.51T + 89T^{2} \) |
| 97 | \( 1 + 16.2T + 97T^{2} \) |
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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
−8.927148941942006475468257359330, −8.028378500570744554438195769805, −6.79787926386540506756196034120, −6.56978481264360987836007181278, −5.85441261264160422923414388709, −4.28846284634197762271215964431, −3.92183914233279468775818914833, −2.46419353973355744717973158392, −1.08366372735667782461405333620, 0,
1.08366372735667782461405333620, 2.46419353973355744717973158392, 3.92183914233279468775818914833, 4.28846284634197762271215964431, 5.85441261264160422923414388709, 6.56978481264360987836007181278, 6.79787926386540506756196034120, 8.028378500570744554438195769805, 8.927148941942006475468257359330
