| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 3.05·11-s − 12-s − 1.64·13-s + 16-s + 2.90·17-s − 18-s + 2.21·19-s + 3.05·22-s − 1.78·23-s + 24-s + 1.64·26-s − 27-s + 5.58·29-s + 0.944·31-s − 32-s + 3.05·33-s − 2.90·34-s + 36-s + 10.7·37-s − 2.21·38-s + 1.64·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s − 0.353·8-s + 0.333·9-s − 0.921·11-s − 0.288·12-s − 0.455·13-s + 0.250·16-s + 0.705·17-s − 0.235·18-s + 0.507·19-s + 0.651·22-s − 0.373·23-s + 0.204·24-s + 0.321·26-s − 0.192·27-s + 1.03·29-s + 0.169·31-s − 0.176·32-s + 0.531·33-s − 0.498·34-s + 0.166·36-s + 1.76·37-s − 0.358·38-s + 0.262·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8708207710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8708207710\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 3.05T + 11T^{2} \) |
| 13 | \( 1 + 1.64T + 13T^{2} \) |
| 17 | \( 1 - 2.90T + 17T^{2} \) |
| 19 | \( 1 - 2.21T + 19T^{2} \) |
| 23 | \( 1 + 1.78T + 23T^{2} \) |
| 29 | \( 1 - 5.58T + 29T^{2} \) |
| 31 | \( 1 - 0.944T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 - 6.67T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 + 5.78T + 53T^{2} \) |
| 59 | \( 1 + 5.97T + 59T^{2} \) |
| 61 | \( 1 - 0.445T + 61T^{2} \) |
| 67 | \( 1 + 1.26T + 67T^{2} \) |
| 71 | \( 1 + 14.8T + 71T^{2} \) |
| 73 | \( 1 + 7.91T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 + 0.874T + 83T^{2} \) |
| 89 | \( 1 - 17.5T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−7.79824236639428501639066594730, −7.46584026460577361912579349452, −6.44626415293166161134764947231, −6.00426015717502655133432570025, −5.08616128600606268845940590193, −4.58034776609250394210554058163, −3.32932012785293717505147357393, −2.64242421486157970215659754554, −1.57716393856419633327759117428, −0.54967463626255961784050727844,
0.54967463626255961784050727844, 1.57716393856419633327759117428, 2.64242421486157970215659754554, 3.32932012785293717505147357393, 4.58034776609250394210554058163, 5.08616128600606268845940590193, 6.00426015717502655133432570025, 6.44626415293166161134764947231, 7.46584026460577361912579349452, 7.79824236639428501639066594730
