| L(s) = 1 | − 2-s + 4-s − 5-s − 2.60·7-s − 8-s + 10-s + 13-s + 2.60·14-s + 16-s − 4.60·17-s + 6.60·19-s − 20-s − 4.60·23-s + 25-s − 26-s − 2.60·28-s + 4.60·29-s + 2·31-s − 32-s + 4.60·34-s + 2.60·35-s + 11.2·37-s − 6.60·38-s + 40-s − 3.21·41-s + 5.21·43-s + 4.60·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s − 0.984·7-s − 0.353·8-s + 0.316·10-s + 0.277·13-s + 0.696·14-s + 0.250·16-s − 1.11·17-s + 1.51·19-s − 0.223·20-s − 0.960·23-s + 0.200·25-s − 0.196·26-s − 0.492·28-s + 0.855·29-s + 0.359·31-s − 0.176·32-s + 0.789·34-s + 0.440·35-s + 1.84·37-s − 1.07·38-s + 0.158·40-s − 0.501·41-s + 0.794·43-s + 0.679·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8608468235\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8608468235\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 2.60T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 + 4.60T + 17T^{2} \) |
| 19 | \( 1 - 6.60T + 19T^{2} \) |
| 23 | \( 1 + 4.60T + 23T^{2} \) |
| 29 | \( 1 - 4.60T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 + 3.21T + 41T^{2} \) |
| 43 | \( 1 - 5.21T + 43T^{2} \) |
| 47 | \( 1 - 9.21T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 9.21T + 59T^{2} \) |
| 61 | \( 1 + 7.21T + 61T^{2} \) |
| 67 | \( 1 + 7.21T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 6.60T + 73T^{2} \) |
| 79 | \( 1 + 1.21T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 3.21T + 89T^{2} \) |
| 97 | \( 1 - 6.60T + 97T^{2} \) |
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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
−9.686343474576278519688191035301, −9.069885176056544170940831504170, −8.164793651194956833843508101638, −7.39804725762929191086830470828, −6.55720106484789861358898283438, −5.83499604682561429171345453924, −4.48122356524826415993480896928, −3.42639705063767266463059399710, −2.44293458422399154911802641782, −0.76094941291504912479203936532,
0.76094941291504912479203936532, 2.44293458422399154911802641782, 3.42639705063767266463059399710, 4.48122356524826415993480896928, 5.83499604682561429171345453924, 6.55720106484789861358898283438, 7.39804725762929191086830470828, 8.164793651194956833843508101638, 9.069885176056544170940831504170, 9.686343474576278519688191035301
