| L(s) = 1 | − 3-s + 5-s − 2·7-s + 9-s + 13-s − 15-s − 8·17-s + 7·19-s + 2·21-s + 25-s − 27-s + 7·29-s − 2·35-s + 10·37-s − 39-s − 3·41-s − 6·43-s + 45-s − 13·47-s − 3·49-s + 8·51-s + 53-s − 7·57-s − 2·59-s + 2·61-s − 2·63-s + 65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.277·13-s − 0.258·15-s − 1.94·17-s + 1.60·19-s + 0.436·21-s + 1/5·25-s − 0.192·27-s + 1.29·29-s − 0.338·35-s + 1.64·37-s − 0.160·39-s − 0.468·41-s − 0.914·43-s + 0.149·45-s − 1.89·47-s − 3/7·49-s + 1.12·51-s + 0.137·53-s − 0.927·57-s − 0.260·59-s + 0.256·61-s − 0.251·63-s + 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.40364553069580, −12.79711832417557, −12.64220594285756, −11.69698364979543, −11.53532230382696, −11.15090737295223, −10.47472992245407, −9.947878548965447, −9.731456012823161, −9.157465978578012, −8.694621959500082, −8.091719128086065, −7.566424489030716, −6.820948313225358, −6.505539991993421, −6.287752586967564, −5.576395398086342, −4.896788708338361, −4.734220785346502, −3.928527111373339, −3.312247840059643, −2.793973939321497, −2.163668751915252, −1.428949433860748, −0.7643705816308611, 0,
0.7643705816308611, 1.428949433860748, 2.163668751915252, 2.793973939321497, 3.312247840059643, 3.928527111373339, 4.734220785346502, 4.896788708338361, 5.576395398086342, 6.287752586967564, 6.505539991993421, 6.820948313225358, 7.566424489030716, 8.091719128086065, 8.694621959500082, 9.157465978578012, 9.731456012823161, 9.947878548965447, 10.47472992245407, 11.15090737295223, 11.53532230382696, 11.69698364979543, 12.64220594285756, 12.79711832417557, 13.40364553069580
