| L(s) = 1 | + 3-s + 0.454·5-s + 9-s − 5.79·11-s − 5.88·13-s + 0.454·15-s + 2.90·17-s + 5.88·19-s − 2.90·23-s − 4.79·25-s + 27-s + 3.54·29-s − 4.33·31-s − 5.79·33-s + 7.70·37-s − 5.88·39-s + 9.58·41-s + 10.7·43-s + 0.454·45-s + 4.90·47-s + 2.90·51-s + 13.1·53-s − 2.63·55-s + 5.88·57-s − 1.79·59-s + 4.67·61-s − 2.67·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.203·5-s + 0.333·9-s − 1.74·11-s − 1.63·13-s + 0.117·15-s + 0.705·17-s + 1.34·19-s − 0.606·23-s − 0.958·25-s + 0.192·27-s + 0.658·29-s − 0.779·31-s − 1.00·33-s + 1.26·37-s − 0.942·39-s + 1.49·41-s + 1.64·43-s + 0.0678·45-s + 0.716·47-s + 0.407·51-s + 1.80·53-s − 0.355·55-s + 0.779·57-s − 0.233·59-s + 0.598·61-s − 0.331·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.036197641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.036197641\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 0.454T + 5T^{2} \) |
| 11 | \( 1 + 5.79T + 11T^{2} \) |
| 13 | \( 1 + 5.88T + 13T^{2} \) |
| 17 | \( 1 - 2.90T + 17T^{2} \) |
| 19 | \( 1 - 5.88T + 19T^{2} \) |
| 23 | \( 1 + 2.90T + 23T^{2} \) |
| 29 | \( 1 - 3.54T + 29T^{2} \) |
| 31 | \( 1 + 4.33T + 31T^{2} \) |
| 37 | \( 1 - 7.70T + 37T^{2} \) |
| 41 | \( 1 - 9.58T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 - 4.90T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 + 1.79T + 59T^{2} \) |
| 61 | \( 1 - 4.67T + 61T^{2} \) |
| 67 | \( 1 - 7.88T + 67T^{2} \) |
| 71 | \( 1 + 0.909T + 71T^{2} \) |
| 73 | \( 1 - 5.20T + 73T^{2} \) |
| 79 | \( 1 + 2.75T + 79T^{2} \) |
| 83 | \( 1 - 9.97T + 83T^{2} \) |
| 89 | \( 1 + 4.90T + 89T^{2} \) |
| 97 | \( 1 + 5.79T + 97T^{2} \) |
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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
−8.042468300297955588302791074540, −7.54845546634070910342607778743, −7.32235997848384174768214349395, −5.88011220658887777047232268547, −5.41572699543653049373586273671, −4.64062627788031100277872170587, −3.68110904109015502664387347680, −2.52004236532419179863488110333, −2.41740677076200962223455075500, −0.74199297978788934558945188527,
0.74199297978788934558945188527, 2.41740677076200962223455075500, 2.52004236532419179863488110333, 3.68110904109015502664387347680, 4.64062627788031100277872170587, 5.41572699543653049373586273671, 5.88011220658887777047232268547, 7.32235997848384174768214349395, 7.54845546634070910342607778743, 8.042468300297955588302791074540
