| L(s) = 1 | + 4·2-s + 2.52·3-s + 16·4-s + 10.1·6-s − 49·7-s + 64·8-s − 236.·9-s − 267.·11-s + 40.4·12-s + 896.·13-s − 196·14-s + 256·16-s + 61.1·17-s − 946.·18-s + 1.62e3·19-s − 123.·21-s − 1.07e3·22-s − 4.28e3·23-s + 161.·24-s + 3.58e3·26-s − 1.21e3·27-s − 784·28-s − 7.42e3·29-s − 8.93e3·31-s + 1.02e3·32-s − 677.·33-s + 244.·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.162·3-s + 0.5·4-s + 0.114·6-s − 0.377·7-s + 0.353·8-s − 0.973·9-s − 0.667·11-s + 0.0811·12-s + 1.47·13-s − 0.267·14-s + 0.250·16-s + 0.0513·17-s − 0.688·18-s + 1.03·19-s − 0.0613·21-s − 0.471·22-s − 1.68·23-s + 0.0573·24-s + 1.03·26-s − 0.320·27-s − 0.188·28-s − 1.63·29-s − 1.67·31-s + 0.176·32-s − 0.108·33-s + 0.0363·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 4T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 49T \) |
| good | 3 | \( 1 - 2.52T + 243T^{2} \) |
| 11 | \( 1 + 267.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 896.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 61.1T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.62e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.28e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.42e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.93e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 640.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.87e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.97e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.06e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.97e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.64e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.39e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.46e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.05e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.40e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.99e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.05e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.03e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.21e5T + 8.58e9T^{2} \) |
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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
−10.39114057147777937668334663494, −9.230204000493647234012080472126, −8.242754111877067258932590688349, −7.29762766101249871028053156706, −5.92579881870772630533061575068, −5.52950674058485534700816339808, −3.89506760959089887745230832371, −3.14175316478817745530629398276, −1.79496734525107940527719963991, 0,
1.79496734525107940527719963991, 3.14175316478817745530629398276, 3.89506760959089887745230832371, 5.52950674058485534700816339808, 5.92579881870772630533061575068, 7.29762766101249871028053156706, 8.242754111877067258932590688349, 9.230204000493647234012080472126, 10.39114057147777937668334663494
