| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s − 2·9-s + 6·11-s + 12-s + 14-s + 16-s + 3·17-s − 2·18-s − 19-s + 21-s + 6·22-s + 3·23-s + 24-s − 5·25-s − 5·27-s + 28-s + 9·29-s + 4·31-s + 32-s + 6·33-s + 3·34-s − 2·36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s + 1.80·11-s + 0.288·12-s + 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.471·18-s − 0.229·19-s + 0.218·21-s + 1.27·22-s + 0.625·23-s + 0.204·24-s − 25-s − 0.962·27-s + 0.188·28-s + 1.67·29-s + 0.718·31-s + 0.176·32-s + 1.04·33-s + 0.514·34-s − 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.719303310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.719303310\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.969576930662872845166803888368, −7.35558189285866919427797765822, −6.25338153432601213583646709472, −6.18304856110129897953773330946, −5.00748366820747128480894903619, −4.36756058739341503144365761002, −3.55502409354175041648709074357, −2.97707225880527315032523526567, −1.96927933577139967417741574766, −1.05236257727154933297058348414,
1.05236257727154933297058348414, 1.96927933577139967417741574766, 2.97707225880527315032523526567, 3.55502409354175041648709074357, 4.36756058739341503144365761002, 5.00748366820747128480894903619, 6.18304856110129897953773330946, 6.25338153432601213583646709472, 7.35558189285866919427797765822, 7.969576930662872845166803888368
