| L(s) = 1 | + 1.21·2-s + 1.31·3-s − 0.525·4-s + 1.59·6-s − 2.90·7-s − 3.06·8-s − 1.28·9-s − 0.214·11-s − 0.688·12-s − 3.52·14-s − 2.67·16-s + 6.42·17-s − 1.55·18-s − 2.21·19-s − 3.80·21-s − 0.260·22-s + 4.68·23-s − 4.02·24-s − 5.61·27-s + 1.52·28-s + 8.70·29-s + 5.59·31-s + 2.88·32-s − 0.280·33-s + 7.80·34-s + 0.673·36-s − 2.28·37-s + ⋯ |
| L(s) = 1 | + 0.858·2-s + 0.756·3-s − 0.262·4-s + 0.649·6-s − 1.09·7-s − 1.08·8-s − 0.426·9-s − 0.0646·11-s − 0.198·12-s − 0.942·14-s − 0.668·16-s + 1.55·17-s − 0.366·18-s − 0.507·19-s − 0.830·21-s − 0.0554·22-s + 0.977·23-s − 0.820·24-s − 1.08·27-s + 0.288·28-s + 1.61·29-s + 1.00·31-s + 0.510·32-s − 0.0489·33-s + 1.33·34-s + 0.112·36-s − 0.374·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.538041088\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.538041088\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.21T + 2T^{2} \) |
| 3 | \( 1 - 1.31T + 3T^{2} \) |
| 7 | \( 1 + 2.90T + 7T^{2} \) |
| 11 | \( 1 + 0.214T + 11T^{2} \) |
| 17 | \( 1 - 6.42T + 17T^{2} \) |
| 19 | \( 1 + 2.21T + 19T^{2} \) |
| 23 | \( 1 - 4.68T + 23T^{2} \) |
| 29 | \( 1 - 8.70T + 29T^{2} \) |
| 31 | \( 1 - 5.59T + 31T^{2} \) |
| 37 | \( 1 + 2.28T + 37T^{2} \) |
| 41 | \( 1 + 3.05T + 41T^{2} \) |
| 43 | \( 1 - 6.36T + 43T^{2} \) |
| 47 | \( 1 + 1.09T + 47T^{2} \) |
| 53 | \( 1 + 6.23T + 53T^{2} \) |
| 59 | \( 1 - 9.26T + 59T^{2} \) |
| 61 | \( 1 + 0.280T + 61T^{2} \) |
| 67 | \( 1 - 7.76T + 67T^{2} \) |
| 71 | \( 1 - 6.08T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 + 9.52T + 83T^{2} \) |
| 89 | \( 1 - 5.61T + 89T^{2} \) |
| 97 | \( 1 - 18.0T + 97T^{2} \) |
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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
−8.515788666886452674183668495883, −7.76700154372687429834130477561, −6.71487164063781241868307298241, −6.12996314920509896819912545903, −5.36784393426314463318916251015, −4.60441166909348913119623895609, −3.56281971992993236525387143163, −3.17615478558072498665926290285, −2.50103964926629481594348090728, −0.74836573588688376833158998024,
0.74836573588688376833158998024, 2.50103964926629481594348090728, 3.17615478558072498665926290285, 3.56281971992993236525387143163, 4.60441166909348913119623895609, 5.36784393426314463318916251015, 6.12996314920509896819912545903, 6.71487164063781241868307298241, 7.76700154372687429834130477561, 8.515788666886452674183668495883
