| L(s) = 1 | − 2.82·3-s + 0.241·5-s + 7-s + 4.96·9-s + 11-s + 13-s − 0.681·15-s − 2.24·17-s − 0.101·19-s − 2.82·21-s + 0.624·23-s − 4.94·25-s − 5.53·27-s − 6.59·29-s − 5.28·31-s − 2.82·33-s + 0.241·35-s + 3.42·37-s − 2.82·39-s + 2.37·41-s + 8.75·43-s + 1.19·45-s + 5.69·47-s + 49-s + 6.32·51-s − 7.36·53-s + 0.241·55-s + ⋯ |
| L(s) = 1 | − 1.62·3-s + 0.108·5-s + 0.377·7-s + 1.65·9-s + 0.301·11-s + 0.277·13-s − 0.175·15-s − 0.543·17-s − 0.0232·19-s − 0.615·21-s + 0.130·23-s − 0.988·25-s − 1.06·27-s − 1.22·29-s − 0.948·31-s − 0.491·33-s + 0.0408·35-s + 0.562·37-s − 0.451·39-s + 0.371·41-s + 1.33·43-s + 0.178·45-s + 0.830·47-s + 0.142·49-s + 0.885·51-s − 1.01·53-s + 0.0325·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 2.82T + 3T^{2} \) |
| 5 | \( 1 - 0.241T + 5T^{2} \) |
| 17 | \( 1 + 2.24T + 17T^{2} \) |
| 19 | \( 1 + 0.101T + 19T^{2} \) |
| 23 | \( 1 - 0.624T + 23T^{2} \) |
| 29 | \( 1 + 6.59T + 29T^{2} \) |
| 31 | \( 1 + 5.28T + 31T^{2} \) |
| 37 | \( 1 - 3.42T + 37T^{2} \) |
| 41 | \( 1 - 2.37T + 41T^{2} \) |
| 43 | \( 1 - 8.75T + 43T^{2} \) |
| 47 | \( 1 - 5.69T + 47T^{2} \) |
| 53 | \( 1 + 7.36T + 53T^{2} \) |
| 59 | \( 1 + 0.198T + 59T^{2} \) |
| 61 | \( 1 + 9.45T + 61T^{2} \) |
| 67 | \( 1 - 2.11T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 + 14.8T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 8.05T + 83T^{2} \) |
| 89 | \( 1 - 4.49T + 89T^{2} \) |
| 97 | \( 1 + 5.42T + 97T^{2} \) |
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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
−7.81998779615105328714469838273, −7.28527770394376317453967347603, −6.35595221192020605817445284295, −5.89546382167176856196860048711, −5.23132265863538209459865687915, −4.42337069578917425965073042464, −3.72767002967493550591693392898, −2.20547642319442244622272480343, −1.20182689743775018969054388270, 0,
1.20182689743775018969054388270, 2.20547642319442244622272480343, 3.72767002967493550591693392898, 4.42337069578917425965073042464, 5.23132265863538209459865687915, 5.89546382167176856196860048711, 6.35595221192020605817445284295, 7.28527770394376317453967347603, 7.81998779615105328714469838273
