| L(s) = 1 | − 2-s + 3-s + 4-s + 3.50·5-s − 6-s − 3.58·7-s − 8-s + 9-s − 3.50·10-s − 1.90·11-s + 12-s − 2.97·13-s + 3.58·14-s + 3.50·15-s + 16-s − 2.57·17-s − 18-s + 19-s + 3.50·20-s − 3.58·21-s + 1.90·22-s + 6.52·23-s − 24-s + 7.27·25-s + 2.97·26-s + 27-s − 3.58·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.56·5-s − 0.408·6-s − 1.35·7-s − 0.353·8-s + 0.333·9-s − 1.10·10-s − 0.573·11-s + 0.288·12-s − 0.825·13-s + 0.957·14-s + 0.904·15-s + 0.250·16-s − 0.625·17-s − 0.235·18-s + 0.229·19-s + 0.783·20-s − 0.781·21-s + 0.405·22-s + 1.36·23-s − 0.204·24-s + 1.45·25-s + 0.583·26-s + 0.192·27-s − 0.677·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| good | 5 | \( 1 - 3.50T + 5T^{2} \) |
| 7 | \( 1 + 3.58T + 7T^{2} \) |
| 11 | \( 1 + 1.90T + 11T^{2} \) |
| 13 | \( 1 + 2.97T + 13T^{2} \) |
| 17 | \( 1 + 2.57T + 17T^{2} \) |
| 23 | \( 1 - 6.52T + 23T^{2} \) |
| 29 | \( 1 + 7.14T + 29T^{2} \) |
| 31 | \( 1 + 7.42T + 31T^{2} \) |
| 37 | \( 1 - 5.70T + 37T^{2} \) |
| 41 | \( 1 + 1.70T + 41T^{2} \) |
| 43 | \( 1 - 5.28T + 43T^{2} \) |
| 47 | \( 1 + 6.76T + 47T^{2} \) |
| 59 | \( 1 - 1.16T + 59T^{2} \) |
| 61 | \( 1 - 14.8T + 61T^{2} \) |
| 67 | \( 1 + 7.07T + 67T^{2} \) |
| 71 | \( 1 - 8.69T + 71T^{2} \) |
| 73 | \( 1 + 2.90T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 + 6.97T + 83T^{2} \) |
| 89 | \( 1 + 3.72T + 89T^{2} \) |
| 97 | \( 1 - 9.35T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.63436238357623844630985807135, −7.03968452341832429018206184205, −6.48627856469913911667332721842, −5.66908293507382265393820286361, −5.10220596745732675278156433910, −3.78159029404592456126133930553, −2.77247694267458742872251867148, −2.44556414542696227446194239184, −1.44356783001275518656885848355, 0,
1.44356783001275518656885848355, 2.44556414542696227446194239184, 2.77247694267458742872251867148, 3.78159029404592456126133930553, 5.10220596745732675278156433910, 5.66908293507382265393820286361, 6.48627856469913911667332721842, 7.03968452341832429018206184205, 7.63436238357623844630985807135
