| L(s) = 1 | − 2.42·5-s + 7-s − 2.55·11-s − 3.09·13-s + 5.52·17-s − 19-s − 3.20·23-s + 0.890·25-s − 8.07·29-s + 5.20·31-s − 2.42·35-s + 9.95·37-s + 2·41-s + 4.55·43-s − 6.72·47-s + 49-s − 8.07·53-s + 6.19·55-s − 13.2·61-s + 7.51·65-s − 9.59·67-s + 6.07·71-s − 14.1·73-s − 2.55·77-s + 11.2·79-s − 12.2·83-s − 13.4·85-s + ⋯ |
| L(s) = 1 | − 1.08·5-s + 0.377·7-s − 0.769·11-s − 0.858·13-s + 1.33·17-s − 0.229·19-s − 0.668·23-s + 0.178·25-s − 1.49·29-s + 0.935·31-s − 0.410·35-s + 1.63·37-s + 0.312·41-s + 0.694·43-s − 0.981·47-s + 0.142·49-s − 1.10·53-s + 0.835·55-s − 1.70·61-s + 0.932·65-s − 1.17·67-s + 0.720·71-s − 1.65·73-s − 0.290·77-s + 1.26·79-s − 1.34·83-s − 1.45·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.044085620\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.044085620\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 2.42T + 5T^{2} \) |
| 11 | \( 1 + 2.55T + 11T^{2} \) |
| 13 | \( 1 + 3.09T + 13T^{2} \) |
| 17 | \( 1 - 5.52T + 17T^{2} \) |
| 23 | \( 1 + 3.20T + 23T^{2} \) |
| 29 | \( 1 + 8.07T + 29T^{2} \) |
| 31 | \( 1 - 5.20T + 31T^{2} \) |
| 37 | \( 1 - 9.95T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4.55T + 43T^{2} \) |
| 47 | \( 1 + 6.72T + 47T^{2} \) |
| 53 | \( 1 + 8.07T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 13.2T + 61T^{2} \) |
| 67 | \( 1 + 9.59T + 67T^{2} \) |
| 71 | \( 1 - 6.07T + 71T^{2} \) |
| 73 | \( 1 + 14.1T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 + 3.10T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
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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
−7.78042062031378141713190260535, −7.40702672061256345362393728802, −6.22924295024071756647702870616, −5.65467456586629149871756104999, −4.76794846443257914856505376847, −4.30915694824932329397670586691, −3.41042451075622054406105696483, −2.73390492438308321940646258871, −1.71971029747140546351139592056, −0.47954583924499177237059039596,
0.47954583924499177237059039596, 1.71971029747140546351139592056, 2.73390492438308321940646258871, 3.41042451075622054406105696483, 4.30915694824932329397670586691, 4.76794846443257914856505376847, 5.65467456586629149871756104999, 6.22924295024071756647702870616, 7.40702672061256345362393728802, 7.78042062031378141713190260535
