| L(s) = 1 | − 1.93·3-s − 5-s + 2.47·7-s + 0.729·9-s + 4.45·11-s − 4.02·13-s + 1.93·15-s + 1.38·17-s − 4.02·19-s − 4.77·21-s − 23-s + 25-s + 4.38·27-s + 2.23·29-s + 5.15·31-s − 8.60·33-s − 2.47·35-s − 7.06·37-s + 7.76·39-s + 5.19·41-s − 9.65·43-s − 0.729·45-s − 7.27·47-s − 0.884·49-s − 2.67·51-s + 0.229·53-s − 4.45·55-s + ⋯ |
| L(s) = 1 | − 1.11·3-s − 0.447·5-s + 0.934·7-s + 0.243·9-s + 1.34·11-s − 1.11·13-s + 0.498·15-s + 0.336·17-s − 0.923·19-s − 1.04·21-s − 0.208·23-s + 0.200·25-s + 0.843·27-s + 0.414·29-s + 0.925·31-s − 1.49·33-s − 0.418·35-s − 1.16·37-s + 1.24·39-s + 0.811·41-s − 1.47·43-s − 0.108·45-s − 1.06·47-s − 0.126·49-s − 0.374·51-s + 0.0315·53-s − 0.600·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 1.93T + 3T^{2} \) |
| 7 | \( 1 - 2.47T + 7T^{2} \) |
| 11 | \( 1 - 4.45T + 11T^{2} \) |
| 13 | \( 1 + 4.02T + 13T^{2} \) |
| 17 | \( 1 - 1.38T + 17T^{2} \) |
| 19 | \( 1 + 4.02T + 19T^{2} \) |
| 29 | \( 1 - 2.23T + 29T^{2} \) |
| 31 | \( 1 - 5.15T + 31T^{2} \) |
| 37 | \( 1 + 7.06T + 37T^{2} \) |
| 41 | \( 1 - 5.19T + 41T^{2} \) |
| 43 | \( 1 + 9.65T + 43T^{2} \) |
| 47 | \( 1 + 7.27T + 47T^{2} \) |
| 53 | \( 1 - 0.229T + 53T^{2} \) |
| 59 | \( 1 + 6.09T + 59T^{2} \) |
| 61 | \( 1 - 8.22T + 61T^{2} \) |
| 67 | \( 1 + 8.66T + 67T^{2} \) |
| 71 | \( 1 + 0.318T + 71T^{2} \) |
| 73 | \( 1 - 5.47T + 73T^{2} \) |
| 79 | \( 1 - 1.75T + 79T^{2} \) |
| 83 | \( 1 - 5.67T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−7.51532623455609868306879028719, −6.59495905303927343856154244085, −6.37822069463982747483907772624, −5.28179934534914695750520335478, −4.81027566171375787456784108570, −4.22034477990801544761770100544, −3.23914779977114554770265087481, −2.05926238088964601160030758814, −1.12573979619492105489395744133, 0,
1.12573979619492105489395744133, 2.05926238088964601160030758814, 3.23914779977114554770265087481, 4.22034477990801544761770100544, 4.81027566171375787456784108570, 5.28179934534914695750520335478, 6.37822069463982747483907772624, 6.59495905303927343856154244085, 7.51532623455609868306879028719
