| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s + 6·11-s − 2·13-s − 14-s + 16-s + 3·17-s + 2·19-s + 6·22-s + 9·23-s − 2·26-s − 28-s + 6·29-s − 4·31-s + 32-s + 3·34-s − 8·37-s + 2·38-s − 9·41-s + 43-s + 6·44-s + 9·46-s − 6·47-s + 49-s − 2·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 1.80·11-s − 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.458·19-s + 1.27·22-s + 1.87·23-s − 0.392·26-s − 0.188·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s + 0.514·34-s − 1.31·37-s + 0.324·38-s − 1.40·41-s + 0.152·43-s + 0.904·44-s + 1.32·46-s − 0.875·47-s + 1/7·49-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.983998796\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.983998796\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.36814463539085487160476263296, −6.82269028685701257432012601355, −6.53855078282022087515484981167, −5.43972791351960786389011586124, −5.06569913354684287261241794557, −4.11329695378760779140257976405, −3.46900486583777337885992832636, −2.89980132426307444411016974360, −1.72727142625998514016519839974, −0.912438303358315894078453570903,
0.912438303358315894078453570903, 1.72727142625998514016519839974, 2.89980132426307444411016974360, 3.46900486583777337885992832636, 4.11329695378760779140257976405, 5.06569913354684287261241794557, 5.43972791351960786389011586124, 6.53855078282022087515484981167, 6.82269028685701257432012601355, 7.36814463539085487160476263296
