L(s) = 1 | − 3.74·5-s − 7-s + 3.65·11-s − 5.54·13-s − 7.20·17-s + 3.30·19-s + 4.98·23-s + 9.05·25-s + 0.490·29-s + 3.89·31-s + 3.74·35-s + 7.89·37-s + 4.76·41-s + 1.60·43-s − 9.63·47-s + 49-s − 8.03·53-s − 13.6·55-s + 1.50·59-s − 2.08·61-s + 20.7·65-s − 3.40·67-s + 10.7·71-s + 9.83·73-s − 3.65·77-s − 3.72·79-s + 11.3·83-s + ⋯ |
L(s) = 1 | − 1.67·5-s − 0.377·7-s + 1.10·11-s − 1.53·13-s − 1.74·17-s + 0.758·19-s + 1.04·23-s + 1.81·25-s + 0.0911·29-s + 0.699·31-s + 0.633·35-s + 1.29·37-s + 0.744·41-s + 0.244·43-s − 1.40·47-s + 0.142·49-s − 1.10·53-s − 1.84·55-s + 0.196·59-s − 0.267·61-s + 2.57·65-s − 0.415·67-s + 1.27·71-s + 1.15·73-s − 0.416·77-s − 0.419·79-s + 1.24·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + 3.74T + 5T^{2} \) |
| 11 | \( 1 - 3.65T + 11T^{2} \) |
| 13 | \( 1 + 5.54T + 13T^{2} \) |
| 17 | \( 1 + 7.20T + 17T^{2} \) |
| 19 | \( 1 - 3.30T + 19T^{2} \) |
| 23 | \( 1 - 4.98T + 23T^{2} \) |
| 29 | \( 1 - 0.490T + 29T^{2} \) |
| 31 | \( 1 - 3.89T + 31T^{2} \) |
| 37 | \( 1 - 7.89T + 37T^{2} \) |
| 41 | \( 1 - 4.76T + 41T^{2} \) |
| 43 | \( 1 - 1.60T + 43T^{2} \) |
| 47 | \( 1 + 9.63T + 47T^{2} \) |
| 53 | \( 1 + 8.03T + 53T^{2} \) |
| 59 | \( 1 - 1.50T + 59T^{2} \) |
| 61 | \( 1 + 2.08T + 61T^{2} \) |
| 67 | \( 1 + 3.40T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 - 9.83T + 73T^{2} \) |
| 79 | \( 1 + 3.72T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 - 7.14T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−7.39377762765123840362827921090, −6.78994158949195607262944988492, −6.32759817256378850141849280004, −4.95149101726470578627842904835, −4.59664297765263944034230612301, −3.90729873446771743250220512164, −3.13057923133597145691614270946, −2.38413483723863661956246056541, −0.958030353699659687071041484775, 0,
0.958030353699659687071041484775, 2.38413483723863661956246056541, 3.13057923133597145691614270946, 3.90729873446771743250220512164, 4.59664297765263944034230612301, 4.95149101726470578627842904835, 6.32759817256378850141849280004, 6.78994158949195607262944988492, 7.39377762765123840362827921090