| L(s) = 1 | + 0.840·2-s − 1.72·3-s − 1.29·4-s + 5-s − 1.45·6-s − 2.76·8-s − 0.00824·9-s + 0.840·10-s − 11-s + 2.23·12-s − 2.36·13-s − 1.72·15-s + 0.264·16-s − 2.68·17-s − 0.00692·18-s − 0.851·19-s − 1.29·20-s − 0.840·22-s − 3.43·23-s + 4.78·24-s + 25-s − 1.98·26-s + 5.20·27-s + 8.27·29-s − 1.45·30-s − 8.64·31-s + 5.75·32-s + ⋯ |
| L(s) = 1 | + 0.593·2-s − 0.998·3-s − 0.647·4-s + 0.447·5-s − 0.593·6-s − 0.978·8-s − 0.00274·9-s + 0.265·10-s − 0.301·11-s + 0.646·12-s − 0.655·13-s − 0.446·15-s + 0.0660·16-s − 0.652·17-s − 0.00163·18-s − 0.195·19-s − 0.289·20-s − 0.179·22-s − 0.715·23-s + 0.977·24-s + 0.200·25-s − 0.389·26-s + 1.00·27-s + 1.53·29-s − 0.265·30-s − 1.55·31-s + 1.01·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8946073248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8946073248\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 0.840T + 2T^{2} \) |
| 3 | \( 1 + 1.72T + 3T^{2} \) |
| 13 | \( 1 + 2.36T + 13T^{2} \) |
| 17 | \( 1 + 2.68T + 17T^{2} \) |
| 19 | \( 1 + 0.851T + 19T^{2} \) |
| 23 | \( 1 + 3.43T + 23T^{2} \) |
| 29 | \( 1 - 8.27T + 29T^{2} \) |
| 31 | \( 1 + 8.64T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + 8.73T + 41T^{2} \) |
| 43 | \( 1 + 2.48T + 43T^{2} \) |
| 47 | \( 1 + 1.23T + 47T^{2} \) |
| 53 | \( 1 - 1.58T + 53T^{2} \) |
| 59 | \( 1 + 1.71T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 2.74T + 67T^{2} \) |
| 71 | \( 1 - 9.43T + 71T^{2} \) |
| 73 | \( 1 + 8.37T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 + 0.868T + 97T^{2} \) |
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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
−8.808544287767929183952384092620, −8.214119036923707465134005582935, −7.03375368555888206667178634973, −6.24949080689266368326040091546, −5.67228695199062796708347489502, −4.94980594152459683294727889583, −4.41942346097425625788412089359, −3.26282351207271964538819880031, −2.21932806215075507141211782282, −0.54778690774648245258095941197,
0.54778690774648245258095941197, 2.21932806215075507141211782282, 3.26282351207271964538819880031, 4.41942346097425625788412089359, 4.94980594152459683294727889583, 5.67228695199062796708347489502, 6.24949080689266368326040091546, 7.03375368555888206667178634973, 8.214119036923707465134005582935, 8.808544287767929183952384092620
