| L(s) = 1 | − 2-s + 3-s + 4-s − 3.93·5-s − 6-s + 7-s − 8-s + 9-s + 3.93·10-s − 5.95·11-s + 12-s − 14-s − 3.93·15-s + 16-s + 5.27·17-s − 18-s + 4.12·19-s − 3.93·20-s + 21-s + 5.95·22-s − 6.25·23-s − 24-s + 10.5·25-s + 27-s + 28-s + 1.98·29-s + 3.93·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.76·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 1.24·10-s − 1.79·11-s + 0.288·12-s − 0.267·14-s − 1.01·15-s + 0.250·16-s + 1.27·17-s − 0.235·18-s + 0.946·19-s − 0.880·20-s + 0.218·21-s + 1.26·22-s − 1.30·23-s − 0.204·24-s + 2.10·25-s + 0.192·27-s + 0.188·28-s + 0.369·29-s + 0.718·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 3.93T + 5T^{2} \) |
| 11 | \( 1 + 5.95T + 11T^{2} \) |
| 17 | \( 1 - 5.27T + 17T^{2} \) |
| 19 | \( 1 - 4.12T + 19T^{2} \) |
| 23 | \( 1 + 6.25T + 23T^{2} \) |
| 29 | \( 1 - 1.98T + 29T^{2} \) |
| 31 | \( 1 - 8.86T + 31T^{2} \) |
| 37 | \( 1 + 4.03T + 37T^{2} \) |
| 41 | \( 1 + 4.03T + 41T^{2} \) |
| 43 | \( 1 + 9.79T + 43T^{2} \) |
| 47 | \( 1 + 0.0488T + 47T^{2} \) |
| 53 | \( 1 + 1.32T + 53T^{2} \) |
| 59 | \( 1 + 1.86T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 + 15.5T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 - 6.79T + 73T^{2} \) |
| 79 | \( 1 - 9.36T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 - 4.89T + 89T^{2} \) |
| 97 | \( 1 - 5.77T + 97T^{2} \) |
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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
−7.81369540076616412884379237409, −7.38416906296382802144053176015, −6.46348177103455490508923107644, −5.26942102506874123073420378864, −4.78433858208047607410855889738, −3.66162449262439096090617573084, −3.19700341638830350679925545603, −2.34910442813073385503722108010, −1.05504702868768977952225465402, 0,
1.05504702868768977952225465402, 2.34910442813073385503722108010, 3.19700341638830350679925545603, 3.66162449262439096090617573084, 4.78433858208047607410855889738, 5.26942102506874123073420378864, 6.46348177103455490508923107644, 7.38416906296382802144053176015, 7.81369540076616412884379237409
