| L(s) = 1 | − 3-s + 2.73·5-s + 7-s + 9-s − 5.46·13-s − 2.73·15-s − 1.26·17-s − 19-s − 21-s + 2.46·25-s − 27-s − 9.66·29-s + 2·31-s + 2.73·35-s − 10·37-s + 5.46·39-s − 8.92·41-s + 4.92·43-s + 2.73·45-s − 11.6·47-s + 49-s + 1.26·51-s − 6.73·53-s + 57-s + 8·59-s − 2·61-s + 63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.22·5-s + 0.377·7-s + 0.333·9-s − 1.51·13-s − 0.705·15-s − 0.307·17-s − 0.229·19-s − 0.218·21-s + 0.492·25-s − 0.192·27-s − 1.79·29-s + 0.359·31-s + 0.461·35-s − 1.64·37-s + 0.874·39-s − 1.39·41-s + 0.751·43-s + 0.407·45-s − 1.70·47-s + 0.142·49-s + 0.177·51-s − 0.924·53-s + 0.132·57-s + 1.04·59-s − 0.256·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 2.73T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 5.46T + 13T^{2} \) |
| 17 | \( 1 + 1.26T + 17T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 9.66T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 + 8.92T + 41T^{2} \) |
| 43 | \( 1 - 4.92T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 + 6.73T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 - 4.73T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 - 4.92T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−8.298109307110716885772338473584, −7.38990802393262026154540378761, −6.74467953239146270778864976808, −5.96050108383078022144209583258, −5.17630396496452879661708451088, −4.78673596878036509175024432651, −3.51852931910054145811193539037, −2.25127277153603816838085760738, −1.68195776420004564847209221213, 0,
1.68195776420004564847209221213, 2.25127277153603816838085760738, 3.51852931910054145811193539037, 4.78673596878036509175024432651, 5.17630396496452879661708451088, 5.96050108383078022144209583258, 6.74467953239146270778864976808, 7.38990802393262026154540378761, 8.298109307110716885772338473584
