| L(s) = 1 | + 2-s − 3-s + 4-s − 1.58·5-s − 6-s + 8-s + 9-s − 1.58·10-s − 11-s − 12-s + 2.82·13-s + 1.58·15-s + 16-s − 4.24·17-s + 18-s + 19-s − 1.58·20-s − 22-s + 3.41·23-s − 24-s − 2.48·25-s + 2.82·26-s − 27-s + 2.17·29-s + 1.58·30-s − 10.6·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.709·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.501·10-s − 0.301·11-s − 0.288·12-s + 0.784·13-s + 0.409·15-s + 0.250·16-s − 1.02·17-s + 0.235·18-s + 0.229·19-s − 0.354·20-s − 0.213·22-s + 0.711·23-s − 0.204·24-s − 0.497·25-s + 0.554·26-s − 0.192·27-s + 0.403·29-s + 0.289·30-s − 1.91·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 1.58T + 5T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 + 4.24T + 17T^{2} \) |
| 23 | \( 1 - 3.41T + 23T^{2} \) |
| 29 | \( 1 - 2.17T + 29T^{2} \) |
| 31 | \( 1 + 10.6T + 31T^{2} \) |
| 37 | \( 1 - 0.242T + 37T^{2} \) |
| 41 | \( 1 + 0.585T + 41T^{2} \) |
| 43 | \( 1 - 6.24T + 43T^{2} \) |
| 47 | \( 1 - 3.17T + 47T^{2} \) |
| 53 | \( 1 - 1.82T + 53T^{2} \) |
| 59 | \( 1 - 6.07T + 59T^{2} \) |
| 61 | \( 1 + 4.82T + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 + 7.89T + 71T^{2} \) |
| 73 | \( 1 + 14.8T + 73T^{2} \) |
| 79 | \( 1 + 0.171T + 79T^{2} \) |
| 83 | \( 1 + 17.1T + 83T^{2} \) |
| 89 | \( 1 - 3.75T + 89T^{2} \) |
| 97 | \( 1 + 5.58T + 97T^{2} \) |
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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
−7.38556353336312878038613447465, −7.20376195431669540433880207836, −6.15748150583024484928937796812, −5.67123041018279430921433965550, −4.80805777510636754975723185551, −4.13543208222462487273065440528, −3.50183375947912428865665553050, −2.48467441877462232304901553534, −1.35857257526470938939943373715, 0,
1.35857257526470938939943373715, 2.48467441877462232304901553534, 3.50183375947912428865665553050, 4.13543208222462487273065440528, 4.80805777510636754975723185551, 5.67123041018279430921433965550, 6.15748150583024484928937796812, 7.20376195431669540433880207836, 7.38556353336312878038613447465
