| L(s) = 1 | + 5-s + 4.60·7-s − 2.60·11-s − 2.60·13-s − 2·17-s − 19-s + 2·23-s + 25-s + 4.60·29-s + 4·31-s + 4.60·35-s + 10.6·37-s + 0.605·41-s + 3.39·43-s − 6·47-s + 14.2·49-s − 2.60·55-s + 9.21·59-s + 7.21·61-s − 2.60·65-s + 4·67-s − 5.21·71-s + 6·73-s − 12·77-s + 8·79-s − 3.21·83-s − 2·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.74·7-s − 0.785·11-s − 0.722·13-s − 0.485·17-s − 0.229·19-s + 0.417·23-s + 0.200·25-s + 0.855·29-s + 0.718·31-s + 0.778·35-s + 1.74·37-s + 0.0945·41-s + 0.517·43-s − 0.875·47-s + 2.03·49-s − 0.351·55-s + 1.19·59-s + 0.923·61-s − 0.323·65-s + 0.488·67-s − 0.618·71-s + 0.702·73-s − 1.36·77-s + 0.900·79-s − 0.352·83-s − 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.418474178\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.418474178\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 - 4.60T + 7T^{2} \) |
| 11 | \( 1 + 2.60T + 11T^{2} \) |
| 13 | \( 1 + 2.60T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 - 4.60T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 - 0.605T + 41T^{2} \) |
| 43 | \( 1 - 3.39T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 9.21T + 59T^{2} \) |
| 61 | \( 1 - 7.21T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 5.21T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 3.21T + 83T^{2} \) |
| 89 | \( 1 - 0.605T + 89T^{2} \) |
| 97 | \( 1 - 9.39T + 97T^{2} \) |
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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
−8.373826473532979353173915277414, −8.017758762469545335207834987866, −7.22995523881623018447391101203, −6.35121972907600575084080843670, −5.35912635621105121152179815685, −4.86301940538579657068269486067, −4.21056087565350576543601853828, −2.70399155726010764220574215707, −2.12808752101794771314031125796, −0.956235109613283930659726593313,
0.956235109613283930659726593313, 2.12808752101794771314031125796, 2.70399155726010764220574215707, 4.21056087565350576543601853828, 4.86301940538579657068269486067, 5.35912635621105121152179815685, 6.35121972907600575084080843670, 7.22995523881623018447391101203, 8.017758762469545335207834987866, 8.373826473532979353173915277414
