L(s) = 1 | + 0.819·2-s + 1.40·3-s − 1.32·4-s + 5-s + 1.14·6-s + 7-s − 2.72·8-s − 1.03·9-s + 0.819·10-s − 4.43·11-s − 1.85·12-s + 0.819·14-s + 1.40·15-s + 0.419·16-s + 6.69·17-s − 0.852·18-s − 1.96·19-s − 1.32·20-s + 1.40·21-s − 3.63·22-s + 5.09·23-s − 3.81·24-s + 25-s − 5.65·27-s − 1.32·28-s − 3.03·29-s + 1.14·30-s + ⋯ |
L(s) = 1 | + 0.579·2-s + 0.808·3-s − 0.664·4-s + 0.447·5-s + 0.468·6-s + 0.377·7-s − 0.964·8-s − 0.346·9-s + 0.259·10-s − 1.33·11-s − 0.536·12-s + 0.219·14-s + 0.361·15-s + 0.104·16-s + 1.62·17-s − 0.200·18-s − 0.449·19-s − 0.296·20-s + 0.305·21-s − 0.775·22-s + 1.06·23-s − 0.779·24-s + 0.200·25-s − 1.08·27-s − 0.250·28-s − 0.564·29-s + 0.209·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5915 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5915 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 0.819T + 2T^{2} \) |
| 3 | \( 1 - 1.40T + 3T^{2} \) |
| 11 | \( 1 + 4.43T + 11T^{2} \) |
| 17 | \( 1 - 6.69T + 17T^{2} \) |
| 19 | \( 1 + 1.96T + 19T^{2} \) |
| 23 | \( 1 - 5.09T + 23T^{2} \) |
| 29 | \( 1 + 3.03T + 29T^{2} \) |
| 31 | \( 1 + 7.03T + 31T^{2} \) |
| 37 | \( 1 + 4.67T + 37T^{2} \) |
| 41 | \( 1 - 1.20T + 41T^{2} \) |
| 43 | \( 1 - 1.63T + 43T^{2} \) |
| 47 | \( 1 - 8.65T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 + 4.96T + 59T^{2} \) |
| 61 | \( 1 + 0.216T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 6.83T + 73T^{2} \) |
| 79 | \( 1 - 1.59T + 79T^{2} \) |
| 83 | \( 1 - 6.15T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−7.84538382899864208263335437734, −7.24348607966245425771975988514, −5.91269974681839235543891304745, −5.50785867846352588800238236155, −4.96160499480693601013153615683, −3.97933360002876818089264392198, −3.13520707085837954800515150810, −2.70624787435248701326030035441, −1.52120488247790183493791305460, 0,
1.52120488247790183493791305460, 2.70624787435248701326030035441, 3.13520707085837954800515150810, 3.97933360002876818089264392198, 4.96160499480693601013153615683, 5.50785867846352588800238236155, 5.91269974681839235543891304745, 7.24348607966245425771975988514, 7.84538382899864208263335437734