| L(s) = 1 | − 1.66·3-s − 3.23·5-s + 4.31·7-s − 0.241·9-s − 0.366·11-s − 0.733·13-s + 5.36·15-s + 1.89·17-s + 0.903·19-s − 7.17·21-s − 4.65·23-s + 5.44·25-s + 5.38·27-s + 6.97·29-s − 10.4·31-s + 0.609·33-s − 13.9·35-s − 8.90·37-s + 1.21·39-s + 5.85·41-s − 8.62·43-s + 0.779·45-s + 6.39·47-s + 11.6·49-s − 3.14·51-s + 7.70·53-s + 1.18·55-s + ⋯ |
| L(s) = 1 | − 0.958·3-s − 1.44·5-s + 1.63·7-s − 0.0803·9-s − 0.110·11-s − 0.203·13-s + 1.38·15-s + 0.459·17-s + 0.207·19-s − 1.56·21-s − 0.969·23-s + 1.08·25-s + 1.03·27-s + 1.29·29-s − 1.88·31-s + 0.106·33-s − 2.35·35-s − 1.46·37-s + 0.194·39-s + 0.914·41-s − 1.31·43-s + 0.116·45-s + 0.932·47-s + 1.66·49-s − 0.440·51-s + 1.05·53-s + 0.159·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 + 1.66T + 3T^{2} \) |
| 5 | \( 1 + 3.23T + 5T^{2} \) |
| 7 | \( 1 - 4.31T + 7T^{2} \) |
| 11 | \( 1 + 0.366T + 11T^{2} \) |
| 13 | \( 1 + 0.733T + 13T^{2} \) |
| 17 | \( 1 - 1.89T + 17T^{2} \) |
| 19 | \( 1 - 0.903T + 19T^{2} \) |
| 23 | \( 1 + 4.65T + 23T^{2} \) |
| 29 | \( 1 - 6.97T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 + 8.90T + 37T^{2} \) |
| 41 | \( 1 - 5.85T + 41T^{2} \) |
| 43 | \( 1 + 8.62T + 43T^{2} \) |
| 47 | \( 1 - 6.39T + 47T^{2} \) |
| 53 | \( 1 - 7.70T + 53T^{2} \) |
| 59 | \( 1 - 2.27T + 59T^{2} \) |
| 61 | \( 1 + 9.01T + 61T^{2} \) |
| 67 | \( 1 - 1.58T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 - 7.65T + 73T^{2} \) |
| 79 | \( 1 - 5.91T + 79T^{2} \) |
| 83 | \( 1 + 16.8T + 83T^{2} \) |
| 89 | \( 1 - 8.32T + 89T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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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
−7.62410539208952991450544220893, −7.26339851809337855047456718237, −6.25421208632295191451619610740, −5.28988640842643418932131486100, −5.01519428303484382426302772466, −4.15683276757335689234513998533, −3.47658823066677281136236675309, −2.20277153085138532416239929507, −1.06589661813065319353713681131, 0,
1.06589661813065319353713681131, 2.20277153085138532416239929507, 3.47658823066677281136236675309, 4.15683276757335689234513998533, 5.01519428303484382426302772466, 5.28988640842643418932131486100, 6.25421208632295191451619610740, 7.26339851809337855047456718237, 7.62410539208952991450544220893
