| L(s) = 1 | + 2.17·2-s − 3-s + 2.73·4-s + 5-s − 2.17·6-s − 3.90·7-s + 1.59·8-s + 9-s + 2.17·10-s + 1.59·11-s − 2.73·12-s − 8.49·14-s − 15-s − 1.99·16-s − 3.76·17-s + 2.17·18-s − 7.91·19-s + 2.73·20-s + 3.90·21-s + 3.46·22-s + 6.22·23-s − 1.59·24-s + 25-s − 27-s − 10.6·28-s − 5.03·29-s − 2.17·30-s + ⋯ |
| L(s) = 1 | + 1.53·2-s − 0.577·3-s + 1.36·4-s + 0.447·5-s − 0.888·6-s − 1.47·7-s + 0.563·8-s + 0.333·9-s + 0.687·10-s + 0.480·11-s − 0.788·12-s − 2.27·14-s − 0.258·15-s − 0.499·16-s − 0.913·17-s + 0.512·18-s − 1.81·19-s + 0.610·20-s + 0.852·21-s + 0.738·22-s + 1.29·23-s − 0.325·24-s + 0.200·25-s − 0.192·27-s − 2.01·28-s − 0.935·29-s − 0.397·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.17T + 2T^{2} \) |
| 7 | \( 1 + 3.90T + 7T^{2} \) |
| 11 | \( 1 - 1.59T + 11T^{2} \) |
| 17 | \( 1 + 3.76T + 17T^{2} \) |
| 19 | \( 1 + 7.91T + 19T^{2} \) |
| 23 | \( 1 - 6.22T + 23T^{2} \) |
| 29 | \( 1 + 5.03T + 29T^{2} \) |
| 31 | \( 1 - 0.184T + 31T^{2} \) |
| 37 | \( 1 + 1.64T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 6.74T + 43T^{2} \) |
| 47 | \( 1 + 6.58T + 47T^{2} \) |
| 53 | \( 1 + 5.51T + 53T^{2} \) |
| 59 | \( 1 - 4.32T + 59T^{2} \) |
| 61 | \( 1 + 9.73T + 61T^{2} \) |
| 67 | \( 1 - 6.16T + 67T^{2} \) |
| 71 | \( 1 - 1.24T + 71T^{2} \) |
| 73 | \( 1 - 2.25T + 73T^{2} \) |
| 79 | \( 1 + 4.33T + 79T^{2} \) |
| 83 | \( 1 + 9.12T + 83T^{2} \) |
| 89 | \( 1 - 9.14T + 89T^{2} \) |
| 97 | \( 1 + 9.58T + 97T^{2} \) |
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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
−8.734202003567200695987835421940, −7.14819797821840618768295021445, −6.44863874882331235172101926594, −6.33436119303283152281556089998, −5.35954297470944437639432176696, −4.55890986501165193047849140364, −3.78809104918744075095532592537, −2.97265227797243777046017230679, −1.95728436263369798318374323925, 0,
1.95728436263369798318374323925, 2.97265227797243777046017230679, 3.78809104918744075095532592537, 4.55890986501165193047849140364, 5.35954297470944437639432176696, 6.33436119303283152281556089998, 6.44863874882331235172101926594, 7.14819797821840618768295021445, 8.734202003567200695987835421940
