| L(s) = 1 | + 1.76·2-s − 0.691·3-s + 1.12·4-s − 1.22·6-s + 4.36·7-s − 1.55·8-s − 2.52·9-s + 5.43·11-s − 0.774·12-s + 7.70·14-s − 4.98·16-s + 3.63·17-s − 4.45·18-s + 1.01·19-s − 3.01·21-s + 9.60·22-s + 3.79·23-s + 1.07·24-s + 3.81·27-s + 4.88·28-s − 1.36·29-s + 0.129·31-s − 5.69·32-s − 3.75·33-s + 6.42·34-s − 2.82·36-s − 5.24·37-s + ⋯ |
| L(s) = 1 | + 1.24·2-s − 0.399·3-s + 0.560·4-s − 0.498·6-s + 1.64·7-s − 0.549·8-s − 0.840·9-s + 1.63·11-s − 0.223·12-s + 2.05·14-s − 1.24·16-s + 0.882·17-s − 1.05·18-s + 0.232·19-s − 0.658·21-s + 2.04·22-s + 0.791·23-s + 0.219·24-s + 0.734·27-s + 0.923·28-s − 0.253·29-s + 0.0232·31-s − 1.00·32-s − 0.654·33-s + 1.10·34-s − 0.471·36-s − 0.861·37-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
| \(L(1)\) |
\(\approx\) |
\(3.894738639\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.894738639\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.76T + 2T^{2} \) |
| 3 | \( 1 + 0.691T + 3T^{2} \) |
| 7 | \( 1 - 4.36T + 7T^{2} \) |
| 11 | \( 1 - 5.43T + 11T^{2} \) |
| 17 | \( 1 - 3.63T + 17T^{2} \) |
| 19 | \( 1 - 1.01T + 19T^{2} \) |
| 23 | \( 1 - 3.79T + 23T^{2} \) |
| 29 | \( 1 + 1.36T + 29T^{2} \) |
| 31 | \( 1 - 0.129T + 31T^{2} \) |
| 37 | \( 1 + 5.24T + 37T^{2} \) |
| 41 | \( 1 + 4.70T + 41T^{2} \) |
| 43 | \( 1 + 0.0572T + 43T^{2} \) |
| 47 | \( 1 + 7.23T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 2.80T + 59T^{2} \) |
| 61 | \( 1 - 5.30T + 61T^{2} \) |
| 67 | \( 1 - 7.94T + 67T^{2} \) |
| 71 | \( 1 - 3.88T + 71T^{2} \) |
| 73 | \( 1 + 4.44T + 73T^{2} \) |
| 79 | \( 1 + 5.43T + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 - 6.84T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−8.559149658621163417494179971734, −7.50371947322414935197503639280, −6.67274049582653176017644037057, −5.96484382982807965414888496684, −5.16963973048512252129297214680, −4.89836813085364997753834608462, −3.89825480010023920304839295744, −3.28900729192210021949757143953, −2.06383828396210886422169191665, −1.01073572790316064877665096833,
1.01073572790316064877665096833, 2.06383828396210886422169191665, 3.28900729192210021949757143953, 3.89825480010023920304839295744, 4.89836813085364997753834608462, 5.16963973048512252129297214680, 5.96484382982807965414888496684, 6.67274049582653176017644037057, 7.50371947322414935197503639280, 8.559149658621163417494179971734
