| L(s) = 1 | + 2.13·2-s + 2.55·4-s + 2.16·7-s + 1.17·8-s + 2.50·11-s − 4.33·13-s + 4.62·14-s − 2.59·16-s + 6.77·17-s + 6.83·19-s + 5.35·22-s + 1.67·23-s − 9.24·26-s + 5.53·28-s − 4.44·29-s + 6.56·31-s − 7.88·32-s + 14.4·34-s + 7.97·37-s + 14.5·38-s − 11.2·41-s − 4.25·43-s + 6.40·44-s + 3.58·46-s − 4.98·47-s − 2.29·49-s − 11.0·52-s + ⋯ |
| L(s) = 1 | + 1.50·2-s + 1.27·4-s + 0.819·7-s + 0.415·8-s + 0.756·11-s − 1.20·13-s + 1.23·14-s − 0.648·16-s + 1.64·17-s + 1.56·19-s + 1.14·22-s + 0.349·23-s − 1.81·26-s + 1.04·28-s − 0.826·29-s + 1.17·31-s − 1.39·32-s + 2.47·34-s + 1.31·37-s + 2.36·38-s − 1.75·41-s − 0.648·43-s + 0.964·44-s + 0.527·46-s − 0.727·47-s − 0.328·49-s − 1.53·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.576312653\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.576312653\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2.13T + 2T^{2} \) |
| 7 | \( 1 - 2.16T + 7T^{2} \) |
| 11 | \( 1 - 2.50T + 11T^{2} \) |
| 13 | \( 1 + 4.33T + 13T^{2} \) |
| 17 | \( 1 - 6.77T + 17T^{2} \) |
| 19 | \( 1 - 6.83T + 19T^{2} \) |
| 23 | \( 1 - 1.67T + 23T^{2} \) |
| 29 | \( 1 + 4.44T + 29T^{2} \) |
| 31 | \( 1 - 6.56T + 31T^{2} \) |
| 37 | \( 1 - 7.97T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 + 4.25T + 43T^{2} \) |
| 47 | \( 1 + 4.98T + 47T^{2} \) |
| 53 | \( 1 - 8.21T + 53T^{2} \) |
| 59 | \( 1 + 3.67T + 59T^{2} \) |
| 61 | \( 1 - 4.93T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 - 2.30T + 71T^{2} \) |
| 73 | \( 1 + 1.11T + 73T^{2} \) |
| 79 | \( 1 - 7.78T + 79T^{2} \) |
| 83 | \( 1 + 9.87T + 83T^{2} \) |
| 89 | \( 1 - 1.24T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−7.83043389479305310468124453487, −7.32985530842890484876694140814, −6.54638208912000248374737982289, −5.70323417920081972796461094275, −5.04728661396744072754254738080, −4.74421109307350660554993153556, −3.63695291972525854165688828589, −3.16598857139219284071935096244, −2.14791532906928456425568894704, −1.06911019066900265686172475930,
1.06911019066900265686172475930, 2.14791532906928456425568894704, 3.16598857139219284071935096244, 3.63695291972525854165688828589, 4.74421109307350660554993153556, 5.04728661396744072754254738080, 5.70323417920081972796461094275, 6.54638208912000248374737982289, 7.32985530842890484876694140814, 7.83043389479305310468124453487
