| L(s) = 1 | − 1.18·3-s − 5-s − 1.59·9-s − 0.230·11-s + 2.27·13-s + 1.18·15-s − 6.53·17-s − 0.260·19-s + 8.87·23-s + 25-s + 5.44·27-s + 5.42·29-s + 4.87·31-s + 0.272·33-s − 1.15·37-s − 2.69·39-s − 4.43·41-s − 4.17·43-s + 1.59·45-s + 0.923·47-s + 7.73·51-s − 2.13·53-s + 0.230·55-s + 0.308·57-s − 5.07·59-s − 8.88·61-s − 2.27·65-s + ⋯ |
| L(s) = 1 | − 0.683·3-s − 0.447·5-s − 0.532·9-s − 0.0694·11-s + 0.630·13-s + 0.305·15-s − 1.58·17-s − 0.0596·19-s + 1.84·23-s + 0.200·25-s + 1.04·27-s + 1.00·29-s + 0.874·31-s + 0.0474·33-s − 0.189·37-s − 0.430·39-s − 0.692·41-s − 0.637·43-s + 0.238·45-s + 0.134·47-s + 1.08·51-s − 0.293·53-s + 0.0310·55-s + 0.0408·57-s − 0.660·59-s − 1.13·61-s − 0.281·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 1.18T + 3T^{2} \) |
| 11 | \( 1 + 0.230T + 11T^{2} \) |
| 13 | \( 1 - 2.27T + 13T^{2} \) |
| 17 | \( 1 + 6.53T + 17T^{2} \) |
| 19 | \( 1 + 0.260T + 19T^{2} \) |
| 23 | \( 1 - 8.87T + 23T^{2} \) |
| 29 | \( 1 - 5.42T + 29T^{2} \) |
| 31 | \( 1 - 4.87T + 31T^{2} \) |
| 37 | \( 1 + 1.15T + 37T^{2} \) |
| 41 | \( 1 + 4.43T + 41T^{2} \) |
| 43 | \( 1 + 4.17T + 43T^{2} \) |
| 47 | \( 1 - 0.923T + 47T^{2} \) |
| 53 | \( 1 + 2.13T + 53T^{2} \) |
| 59 | \( 1 + 5.07T + 59T^{2} \) |
| 61 | \( 1 + 8.88T + 61T^{2} \) |
| 67 | \( 1 + 8.15T + 67T^{2} \) |
| 71 | \( 1 - 9.06T + 71T^{2} \) |
| 73 | \( 1 + 6.00T + 73T^{2} \) |
| 79 | \( 1 + 0.112T + 79T^{2} \) |
| 83 | \( 1 - 8.72T + 83T^{2} \) |
| 89 | \( 1 + 15.9T + 89T^{2} \) |
| 97 | \( 1 + 4.29T + 97T^{2} \) |
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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
−8.340025677621758299829978210546, −7.16772147569999064298209711749, −6.59050551580783476975298191236, −5.99320902005876404141593064329, −4.92620362068911681919703936604, −4.56838597758166729612011108970, −3.36251426441858865369844427257, −2.61114022279491108484573982544, −1.19111182230048337096050547225, 0,
1.19111182230048337096050547225, 2.61114022279491108484573982544, 3.36251426441858865369844427257, 4.56838597758166729612011108970, 4.92620362068911681919703936604, 5.99320902005876404141593064329, 6.59050551580783476975298191236, 7.16772147569999064298209711749, 8.340025677621758299829978210546
