| L(s) = 1 | − 0.324·3-s − 5-s − 1.35·7-s − 2.89·9-s − 2.28·11-s − 1.54·13-s + 0.324·15-s + 7.82·17-s + 4.53·19-s + 0.438·21-s + 0.446·23-s + 25-s + 1.91·27-s + 2.54·29-s − 4.39·31-s + 0.740·33-s + 1.35·35-s − 5.52·37-s + 0.501·39-s + 2.55·41-s − 1.78·43-s + 2.89·45-s − 5.32·47-s − 5.16·49-s − 2.53·51-s + 0.0646·53-s + 2.28·55-s + ⋯ |
| L(s) = 1 | − 0.187·3-s − 0.447·5-s − 0.511·7-s − 0.964·9-s − 0.688·11-s − 0.429·13-s + 0.0836·15-s + 1.89·17-s + 1.04·19-s + 0.0956·21-s + 0.0930·23-s + 0.200·25-s + 0.367·27-s + 0.473·29-s − 0.789·31-s + 0.128·33-s + 0.228·35-s − 0.908·37-s + 0.0803·39-s + 0.398·41-s − 0.272·43-s + 0.431·45-s − 0.776·47-s − 0.738·49-s − 0.355·51-s + 0.00887·53-s + 0.307·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 + 0.324T + 3T^{2} \) |
| 7 | \( 1 + 1.35T + 7T^{2} \) |
| 11 | \( 1 + 2.28T + 11T^{2} \) |
| 13 | \( 1 + 1.54T + 13T^{2} \) |
| 17 | \( 1 - 7.82T + 17T^{2} \) |
| 19 | \( 1 - 4.53T + 19T^{2} \) |
| 23 | \( 1 - 0.446T + 23T^{2} \) |
| 29 | \( 1 - 2.54T + 29T^{2} \) |
| 31 | \( 1 + 4.39T + 31T^{2} \) |
| 37 | \( 1 + 5.52T + 37T^{2} \) |
| 41 | \( 1 - 2.55T + 41T^{2} \) |
| 43 | \( 1 + 1.78T + 43T^{2} \) |
| 47 | \( 1 + 5.32T + 47T^{2} \) |
| 53 | \( 1 - 0.0646T + 53T^{2} \) |
| 59 | \( 1 - 8.28T + 59T^{2} \) |
| 61 | \( 1 + 4.00T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 + 8.01T + 73T^{2} \) |
| 79 | \( 1 - 2.47T + 79T^{2} \) |
| 83 | \( 1 + 4.09T + 83T^{2} \) |
| 89 | \( 1 - 8.32T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−7.61490899011694972223131209291, −6.85056735188474463571223633822, −6.01179662350083819912504149351, −5.28748555414138424350977001686, −4.98871468672910242057076605109, −3.56238611925007658525348105200, −3.28319358487144784977064605283, −2.40690447128626917001964840564, −1.05237591476129300001690471069, 0,
1.05237591476129300001690471069, 2.40690447128626917001964840564, 3.28319358487144784977064605283, 3.56238611925007658525348105200, 4.98871468672910242057076605109, 5.28748555414138424350977001686, 6.01179662350083819912504149351, 6.85056735188474463571223633822, 7.61490899011694972223131209291
