| L(s) = 1 | + 2-s + 4-s − 3·5-s + 7-s + 8-s − 3·10-s − 11-s − 13-s + 14-s + 16-s − 5·17-s − 19-s − 3·20-s − 22-s + 23-s + 4·25-s − 26-s + 28-s − 7·29-s − 2·31-s + 32-s − 5·34-s − 3·35-s + 37-s − 38-s − 3·40-s − 6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.377·7-s + 0.353·8-s − 0.948·10-s − 0.301·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 1.21·17-s − 0.229·19-s − 0.670·20-s − 0.213·22-s + 0.208·23-s + 4/5·25-s − 0.196·26-s + 0.188·28-s − 1.29·29-s − 0.359·31-s + 0.176·32-s − 0.857·34-s − 0.507·35-s + 0.164·37-s − 0.162·38-s − 0.474·40-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−8.753348500037287827964908129272, −8.087399671123924969643801147843, −7.31743211521265601126957812633, −6.69061749873854041708189827769, −5.51931339869349074764387757967, −4.66162486191056383154167071598, −4.01680586816339694050615224274, −3.11346507231815676323880972931, −1.89913794199562214175475004949, 0,
1.89913794199562214175475004949, 3.11346507231815676323880972931, 4.01680586816339694050615224274, 4.66162486191056383154167071598, 5.51931339869349074764387757967, 6.69061749873854041708189827769, 7.31743211521265601126957812633, 8.087399671123924969643801147843, 8.753348500037287827964908129272
