| L(s) = 1 | + 3-s + 1.71·5-s + 0.210·7-s + 9-s − 2.37·11-s − 3.22·13-s + 1.71·15-s − 1.22·17-s − 7.29·19-s + 0.210·21-s + 23-s − 2.04·25-s + 27-s + 29-s + 3.68·31-s − 2.37·33-s + 0.361·35-s + 7.65·37-s − 3.22·39-s + 2.82·41-s + 9.02·43-s + 1.71·45-s − 5.43·47-s − 6.95·49-s − 1.22·51-s − 2.49·53-s − 4.07·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.768·5-s + 0.0793·7-s + 0.333·9-s − 0.715·11-s − 0.894·13-s + 0.443·15-s − 0.298·17-s − 1.67·19-s + 0.0458·21-s + 0.208·23-s − 0.408·25-s + 0.192·27-s + 0.185·29-s + 0.662·31-s − 0.413·33-s + 0.0610·35-s + 1.25·37-s − 0.516·39-s + 0.441·41-s + 1.37·43-s + 0.256·45-s − 0.792·47-s − 0.993·49-s − 0.172·51-s − 0.342·53-s − 0.550·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 - 1.71T + 5T^{2} \) |
| 7 | \( 1 - 0.210T + 7T^{2} \) |
| 11 | \( 1 + 2.37T + 11T^{2} \) |
| 13 | \( 1 + 3.22T + 13T^{2} \) |
| 17 | \( 1 + 1.22T + 17T^{2} \) |
| 19 | \( 1 + 7.29T + 19T^{2} \) |
| 31 | \( 1 - 3.68T + 31T^{2} \) |
| 37 | \( 1 - 7.65T + 37T^{2} \) |
| 41 | \( 1 - 2.82T + 41T^{2} \) |
| 43 | \( 1 - 9.02T + 43T^{2} \) |
| 47 | \( 1 + 5.43T + 47T^{2} \) |
| 53 | \( 1 + 2.49T + 53T^{2} \) |
| 59 | \( 1 + 3.55T + 59T^{2} \) |
| 61 | \( 1 + 7.27T + 61T^{2} \) |
| 67 | \( 1 - 9.40T + 67T^{2} \) |
| 71 | \( 1 + 7.53T + 71T^{2} \) |
| 73 | \( 1 - 3.75T + 73T^{2} \) |
| 79 | \( 1 - 0.807T + 79T^{2} \) |
| 83 | \( 1 - 5.58T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−7.71780819858286144563731701799, −6.69615536238952654517844446784, −6.22526858609929570752577706569, −5.36773294859359879627230164326, −4.60927789440921469308837766118, −4.01923417417396987408866195168, −2.68732456205018528481332524017, −2.47800045106545242599413016678, −1.48929284275036063428538481384, 0,
1.48929284275036063428538481384, 2.47800045106545242599413016678, 2.68732456205018528481332524017, 4.01923417417396987408866195168, 4.60927789440921469308837766118, 5.36773294859359879627230164326, 6.22526858609929570752577706569, 6.69615536238952654517844446784, 7.71780819858286144563731701799
