| L(s) = 1 | + 2.94·3-s + 3.18·5-s − 1.60·7-s + 5.69·9-s − 4.47·11-s − 5.83·13-s + 9.40·15-s − 6.89·17-s − 0.522·19-s − 4.73·21-s − 8.32·23-s + 5.16·25-s + 7.96·27-s + 6.94·29-s − 5.79·31-s − 13.1·33-s − 5.12·35-s − 1.08·37-s − 17.1·39-s − 1.51·41-s + 0.793·43-s + 18.1·45-s − 11.8·47-s − 4.41·49-s − 20.3·51-s − 4.60·53-s − 14.2·55-s + ⋯ |
| L(s) = 1 | + 1.70·3-s + 1.42·5-s − 0.607·7-s + 1.89·9-s − 1.34·11-s − 1.61·13-s + 2.42·15-s − 1.67·17-s − 0.119·19-s − 1.03·21-s − 1.73·23-s + 1.03·25-s + 1.53·27-s + 1.28·29-s − 1.04·31-s − 2.29·33-s − 0.865·35-s − 0.178·37-s − 2.75·39-s − 0.236·41-s + 0.120·43-s + 2.70·45-s − 1.73·47-s − 0.631·49-s − 2.84·51-s − 0.632·53-s − 1.92·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 2.94T + 3T^{2} \) |
| 5 | \( 1 - 3.18T + 5T^{2} \) |
| 7 | \( 1 + 1.60T + 7T^{2} \) |
| 11 | \( 1 + 4.47T + 11T^{2} \) |
| 13 | \( 1 + 5.83T + 13T^{2} \) |
| 17 | \( 1 + 6.89T + 17T^{2} \) |
| 19 | \( 1 + 0.522T + 19T^{2} \) |
| 23 | \( 1 + 8.32T + 23T^{2} \) |
| 29 | \( 1 - 6.94T + 29T^{2} \) |
| 31 | \( 1 + 5.79T + 31T^{2} \) |
| 37 | \( 1 + 1.08T + 37T^{2} \) |
| 41 | \( 1 + 1.51T + 41T^{2} \) |
| 43 | \( 1 - 0.793T + 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 + 4.60T + 53T^{2} \) |
| 59 | \( 1 - 9.50T + 59T^{2} \) |
| 61 | \( 1 - 8.00T + 61T^{2} \) |
| 67 | \( 1 + 6.48T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 - 14.4T + 73T^{2} \) |
| 79 | \( 1 + 2.28T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 + 1.44T + 89T^{2} \) |
| 97 | \( 1 + 3.31T + 97T^{2} \) |
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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
−7.66480899500383607522672735491, −6.79860061582143006586226049905, −6.32924096265946770703235278266, −5.22714776280705358501582853628, −4.70886391329875348166681498679, −3.68233677448613638225207864076, −2.77700495871297836822928166097, −2.20645397686854184798726191435, −2.00732191698545814211483764038, 0,
2.00732191698545814211483764038, 2.20645397686854184798726191435, 2.77700495871297836822928166097, 3.68233677448613638225207864076, 4.70886391329875348166681498679, 5.22714776280705358501582853628, 6.32924096265946770703235278266, 6.79860061582143006586226049905, 7.66480899500383607522672735491
