| L(s) = 1 | + 5.48·3-s + 188.·7-s − 212.·9-s − 501.·11-s − 1.06e3·13-s + 29.5·17-s + 1.57e3·19-s + 1.03e3·21-s + 1.29e3·23-s − 2.50e3·27-s − 3.58e3·29-s − 3.52e3·31-s − 2.75e3·33-s − 8.41e3·37-s − 5.82e3·39-s + 7.01e3·41-s − 2.26e4·43-s + 3.50e3·47-s + 1.89e4·49-s + 162.·51-s − 2.73e4·53-s + 8.66e3·57-s + 7.92e3·59-s − 7.02e3·61-s − 4.02e4·63-s − 1.76e4·67-s + 7.11e3·69-s + ⋯ |
| L(s) = 1 | + 0.352·3-s + 1.45·7-s − 0.875·9-s − 1.25·11-s − 1.74·13-s + 0.0248·17-s + 1.00·19-s + 0.513·21-s + 0.510·23-s − 0.660·27-s − 0.791·29-s − 0.659·31-s − 0.440·33-s − 1.01·37-s − 0.613·39-s + 0.651·41-s − 1.87·43-s + 0.231·47-s + 1.12·49-s + 0.00874·51-s − 1.33·53-s + 0.353·57-s + 0.296·59-s − 0.241·61-s − 1.27·63-s − 0.479·67-s + 0.179·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 5.48T + 243T^{2} \) |
| 7 | \( 1 - 188.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 501.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.06e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 29.5T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.57e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.29e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.58e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.52e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.41e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.01e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.26e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 3.50e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.73e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 7.92e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 7.02e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.76e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.34e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.99e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.33e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.84e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.39e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.10e5T + 8.58e9T^{2} \) |
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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
−11.18910866850743050018944582390, −10.13495730791107210577396448082, −8.985026289184460882430243629619, −7.916951376649847115543613895512, −7.38379614581742832381396856189, −5.40648516554521891406361193444, −4.89645388552753982174570349567, −3.03248010685720210529637750964, −1.92997552332975060274339598469, 0,
1.92997552332975060274339598469, 3.03248010685720210529637750964, 4.89645388552753982174570349567, 5.40648516554521891406361193444, 7.38379614581742832381396856189, 7.916951376649847115543613895512, 8.985026289184460882430243629619, 10.13495730791107210577396448082, 11.18910866850743050018944582390
