| L(s) = 1 | − 3-s − 5-s − 7-s − 2·9-s − 3·11-s + 4·13-s + 15-s + 21-s + 25-s + 5·27-s − 2·31-s + 3·33-s + 35-s − 37-s − 4·39-s − 3·41-s + 2·43-s + 2·45-s + 3·47-s − 6·49-s + 9·53-s + 3·55-s + 2·61-s + 2·63-s − 4·65-s + 4·67-s − 15·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s − 2/3·9-s − 0.904·11-s + 1.10·13-s + 0.258·15-s + 0.218·21-s + 1/5·25-s + 0.962·27-s − 0.359·31-s + 0.522·33-s + 0.169·35-s − 0.164·37-s − 0.640·39-s − 0.468·41-s + 0.304·43-s + 0.298·45-s + 0.437·47-s − 6/7·49-s + 1.23·53-s + 0.404·55-s + 0.256·61-s + 0.251·63-s − 0.496·65-s + 0.488·67-s − 1.78·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 267140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 267140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.08407761250571, −12.45111872509364, −12.14111412246445, −11.48824887870282, −11.26202146531931, −10.82579160282917, −10.19995706259707, −10.11911369071362, −9.086139399918818, −8.925685488442119, −8.325640947344178, −7.960596608604999, −7.367781436247095, −6.857739110639926, −6.347049788911317, −5.825793331822417, −5.512087810800233, −4.953588286510042, −4.377277994784376, −3.762125776358129, −3.257787455261299, −2.787926493785209, −2.132627136677170, −1.315603257733466, −0.6112057937900737, 0,
0.6112057937900737, 1.315603257733466, 2.132627136677170, 2.787926493785209, 3.257787455261299, 3.762125776358129, 4.377277994784376, 4.953588286510042, 5.512087810800233, 5.825793331822417, 6.347049788911317, 6.857739110639926, 7.367781436247095, 7.960596608604999, 8.325640947344178, 8.925685488442119, 9.086139399918818, 10.11911369071362, 10.19995706259707, 10.82579160282917, 11.26202146531931, 11.48824887870282, 12.14111412246445, 12.45111872509364, 13.08407761250571
