| L(s) = 1 | + 1.41·3-s + 0.0681·5-s + 2.73·7-s − 0.999·9-s − 2.73·11-s + 0.0352·13-s + 0.0963·15-s − 1.55·17-s − 2.09·19-s + 3.86·21-s − 4.99·25-s − 5.65·27-s − 1.93·29-s − 9.32·31-s − 3.86·33-s + 0.186·35-s − 9.08·37-s + 0.0498·39-s − 4.46·41-s − 2.02·43-s − 0.0681·45-s − 0.944·47-s + 0.464·49-s − 2.19·51-s + 12.4·53-s − 0.186·55-s − 2.96·57-s + ⋯ |
| L(s) = 1 | + 0.816·3-s + 0.0304·5-s + 1.03·7-s − 0.333·9-s − 0.823·11-s + 0.00978·13-s + 0.0248·15-s − 0.376·17-s − 0.480·19-s + 0.843·21-s − 0.999·25-s − 1.08·27-s − 0.360·29-s − 1.67·31-s − 0.672·33-s + 0.0314·35-s − 1.49·37-s + 0.00798·39-s − 0.697·41-s − 0.309·43-s − 0.0101·45-s − 0.137·47-s + 0.0663·49-s − 0.307·51-s + 1.71·53-s − 0.0251·55-s − 0.392·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 5 | \( 1 - 0.0681T + 5T^{2} \) |
| 7 | \( 1 - 2.73T + 7T^{2} \) |
| 11 | \( 1 + 2.73T + 11T^{2} \) |
| 13 | \( 1 - 0.0352T + 13T^{2} \) |
| 17 | \( 1 + 1.55T + 17T^{2} \) |
| 19 | \( 1 + 2.09T + 19T^{2} \) |
| 29 | \( 1 + 1.93T + 29T^{2} \) |
| 31 | \( 1 + 9.32T + 31T^{2} \) |
| 37 | \( 1 + 9.08T + 37T^{2} \) |
| 41 | \( 1 + 4.46T + 41T^{2} \) |
| 43 | \( 1 + 2.02T + 43T^{2} \) |
| 47 | \( 1 + 0.944T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 + 9.47T + 59T^{2} \) |
| 61 | \( 1 - 6.81T + 61T^{2} \) |
| 67 | \( 1 - 3.19T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 + 7.16T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−8.168197067234725666583105825211, −7.51961378257545500581861588905, −6.73786663770187935745088053214, −5.53564401708726523958107772422, −5.21275530275615655921898197779, −4.08628630337919275359037980560, −3.39495664764832509840552415114, −2.28863248636195656842341502853, −1.79949217230409809256128206539, 0,
1.79949217230409809256128206539, 2.28863248636195656842341502853, 3.39495664764832509840552415114, 4.08628630337919275359037980560, 5.21275530275615655921898197779, 5.53564401708726523958107772422, 6.73786663770187935745088053214, 7.51961378257545500581861588905, 8.168197067234725666583105825211
