| L(s) = 1 | − 3·3-s + 7·7-s + 9·9-s − 54·11-s + 55·13-s − 18·17-s − 25·19-s − 21·21-s + 18·23-s − 27·27-s − 54·29-s − 271·31-s + 162·33-s − 314·37-s − 165·39-s − 360·41-s + 163·43-s − 522·47-s − 294·49-s + 54·51-s + 36·53-s + 75·57-s + 126·59-s + 47·61-s + 63·63-s + 343·67-s − 54·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.48·11-s + 1.17·13-s − 0.256·17-s − 0.301·19-s − 0.218·21-s + 0.163·23-s − 0.192·27-s − 0.345·29-s − 1.57·31-s + 0.854·33-s − 1.39·37-s − 0.677·39-s − 1.37·41-s + 0.578·43-s − 1.62·47-s − 6/7·49-s + 0.148·51-s + 0.0933·53-s + 0.174·57-s + 0.278·59-s + 0.0986·61-s + 0.125·63-s + 0.625·67-s − 0.0942·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - p T + p^{3} T^{2} \) |
| 11 | \( 1 + 54 T + p^{3} T^{2} \) |
| 13 | \( 1 - 55 T + p^{3} T^{2} \) |
| 17 | \( 1 + 18 T + p^{3} T^{2} \) |
| 19 | \( 1 + 25 T + p^{3} T^{2} \) |
| 23 | \( 1 - 18 T + p^{3} T^{2} \) |
| 29 | \( 1 + 54 T + p^{3} T^{2} \) |
| 31 | \( 1 + 271 T + p^{3} T^{2} \) |
| 37 | \( 1 + 314 T + p^{3} T^{2} \) |
| 41 | \( 1 + 360 T + p^{3} T^{2} \) |
| 43 | \( 1 - 163 T + p^{3} T^{2} \) |
| 47 | \( 1 + 522 T + p^{3} T^{2} \) |
| 53 | \( 1 - 36 T + p^{3} T^{2} \) |
| 59 | \( 1 - 126 T + p^{3} T^{2} \) |
| 61 | \( 1 - 47 T + p^{3} T^{2} \) |
| 67 | \( 1 - 343 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1080 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1054 T + p^{3} T^{2} \) |
| 79 | \( 1 + 568 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1422 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1440 T + p^{3} T^{2} \) |
| 97 | \( 1 - 439 T + p^{3} T^{2} \) |
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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
−10.88077040247756589829471580466, −10.18263077548424889544170126357, −8.841016450479790414705282170216, −7.960610173605848451664836178499, −6.87362623219492454821786080292, −5.70309854413014909151053217901, −4.89302660743230191859768776140, −3.46576163101441645613342787643, −1.78586070696064921915361779673, 0,
1.78586070696064921915361779673, 3.46576163101441645613342787643, 4.89302660743230191859768776140, 5.70309854413014909151053217901, 6.87362623219492454821786080292, 7.960610173605848451664836178499, 8.841016450479790414705282170216, 10.18263077548424889544170126357, 10.88077040247756589829471580466
