| L(s) = 1 | − 1.95·2-s − 2.06·3-s + 1.81·4-s + 4.03·6-s − 1.53·7-s + 0.360·8-s + 1.27·9-s + 5.24·11-s − 3.75·12-s + 3.00·14-s − 4.33·16-s − 7.38·17-s − 2.48·18-s − 1.32·19-s + 3.18·21-s − 10.2·22-s + 5.99·23-s − 0.744·24-s + 3.56·27-s − 2.79·28-s + 8.96·29-s − 0.543·31-s + 7.74·32-s − 10.8·33-s + 14.4·34-s + 2.31·36-s − 6.42·37-s + ⋯ |
| L(s) = 1 | − 1.38·2-s − 1.19·3-s + 0.907·4-s + 1.64·6-s − 0.581·7-s + 0.127·8-s + 0.424·9-s + 1.58·11-s − 1.08·12-s + 0.803·14-s − 1.08·16-s − 1.79·17-s − 0.586·18-s − 0.303·19-s + 0.694·21-s − 2.18·22-s + 1.24·23-s − 0.152·24-s + 0.686·27-s − 0.527·28-s + 1.66·29-s − 0.0975·31-s + 1.36·32-s − 1.88·33-s + 2.47·34-s + 0.385·36-s − 1.05·37-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.95T + 2T^{2} \) |
| 3 | \( 1 + 2.06T + 3T^{2} \) |
| 7 | \( 1 + 1.53T + 7T^{2} \) |
| 11 | \( 1 - 5.24T + 11T^{2} \) |
| 17 | \( 1 + 7.38T + 17T^{2} \) |
| 19 | \( 1 + 1.32T + 19T^{2} \) |
| 23 | \( 1 - 5.99T + 23T^{2} \) |
| 29 | \( 1 - 8.96T + 29T^{2} \) |
| 31 | \( 1 + 0.543T + 31T^{2} \) |
| 37 | \( 1 + 6.42T + 37T^{2} \) |
| 41 | \( 1 + 9.60T + 41T^{2} \) |
| 43 | \( 1 + 6.05T + 43T^{2} \) |
| 47 | \( 1 + 5.26T + 47T^{2} \) |
| 53 | \( 1 - 5.94T + 53T^{2} \) |
| 59 | \( 1 + 4.08T + 59T^{2} \) |
| 61 | \( 1 - 3.53T + 61T^{2} \) |
| 67 | \( 1 - 4.74T + 67T^{2} \) |
| 71 | \( 1 - 3.29T + 71T^{2} \) |
| 73 | \( 1 + 3.10T + 73T^{2} \) |
| 79 | \( 1 - 8.09T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 - 17.0T + 89T^{2} \) |
| 97 | \( 1 + 1.35T + 97T^{2} \) |
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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
−8.434713312052809931299929943688, −6.95263204744019706436334379011, −6.73476404160545942483237886158, −6.31808277214391311818628376467, −5.01093862179243984175837781022, −4.47325364903761135155583593786, −3.29341663562465595345948126101, −1.95671637115994223274549358720, −0.966490478235978705045366004955, 0,
0.966490478235978705045366004955, 1.95671637115994223274549358720, 3.29341663562465595345948126101, 4.47325364903761135155583593786, 5.01093862179243984175837781022, 6.31808277214391311818628376467, 6.73476404160545942483237886158, 6.95263204744019706436334379011, 8.434713312052809931299929943688
