| L(s) = 1 | − 2.79·2-s + 5.79·4-s + 3.71·7-s − 10.6·8-s + 5.02·11-s + 13-s − 10.3·14-s + 18.0·16-s + 2.55·17-s + 6.79·19-s − 14.0·22-s + 2.23·23-s − 2.79·26-s + 21.5·28-s + 8.14·29-s − 4.51·31-s − 29.1·32-s − 7.14·34-s + 0.630·37-s − 18.9·38-s + 2.23·41-s + 2.79·43-s + 29.1·44-s − 6.22·46-s + 2.79·47-s + 6.79·49-s + 5.79·52-s + ⋯ |
| L(s) = 1 | − 1.97·2-s + 2.89·4-s + 1.40·7-s − 3.75·8-s + 1.51·11-s + 0.277·13-s − 2.77·14-s + 4.50·16-s + 0.620·17-s + 1.55·19-s − 2.99·22-s + 0.465·23-s − 0.547·26-s + 4.07·28-s + 1.51·29-s − 0.810·31-s − 5.14·32-s − 1.22·34-s + 0.103·37-s − 3.07·38-s + 0.348·41-s + 0.426·43-s + 4.39·44-s − 0.918·46-s + 0.407·47-s + 0.971·49-s + 0.804·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.200415279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.200415279\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 2.79T + 2T^{2} \) |
| 7 | \( 1 - 3.71T + 7T^{2} \) |
| 11 | \( 1 - 5.02T + 11T^{2} \) |
| 17 | \( 1 - 2.55T + 17T^{2} \) |
| 19 | \( 1 - 6.79T + 19T^{2} \) |
| 23 | \( 1 - 2.23T + 23T^{2} \) |
| 29 | \( 1 - 8.14T + 29T^{2} \) |
| 31 | \( 1 + 4.51T + 31T^{2} \) |
| 37 | \( 1 - 0.630T + 37T^{2} \) |
| 41 | \( 1 - 2.23T + 41T^{2} \) |
| 43 | \( 1 - 2.79T + 43T^{2} \) |
| 47 | \( 1 - 2.79T + 47T^{2} \) |
| 53 | \( 1 + 5.91T + 53T^{2} \) |
| 59 | \( 1 - 8.37T + 59T^{2} \) |
| 61 | \( 1 + 8.02T + 61T^{2} \) |
| 67 | \( 1 + 2.51T + 67T^{2} \) |
| 71 | \( 1 + 8.28T + 71T^{2} \) |
| 73 | \( 1 - 14.3T + 73T^{2} \) |
| 79 | \( 1 + 8.39T + 79T^{2} \) |
| 83 | \( 1 - 7.90T + 83T^{2} \) |
| 89 | \( 1 - 4.92T + 89T^{2} \) |
| 97 | \( 1 + 3.42T + 97T^{2} \) |
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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
−8.842961957931513003541840665019, −8.078062596514332898735459468989, −7.55048079628055525542914914432, −6.85532629679846913762832002954, −6.04464842421340018568519026100, −5.11665800812227102224222835422, −3.72054984379743802638584176814, −2.66344735774223958763629394933, −1.39877020689670312464136117846, −1.11288436681930434807593850766,
1.11288436681930434807593850766, 1.39877020689670312464136117846, 2.66344735774223958763629394933, 3.72054984379743802638584176814, 5.11665800812227102224222835422, 6.04464842421340018568519026100, 6.85532629679846913762832002954, 7.55048079628055525542914914432, 8.078062596514332898735459468989, 8.842961957931513003541840665019
