| L(s) = 1 | − 5-s − 4.60·7-s + 11-s − 4.60·13-s − 6.60·17-s + 7.21·19-s + 25-s − 8·29-s − 9.21·31-s + 4.60·35-s − 3.21·37-s − 8·41-s + 3.39·43-s − 5.21·47-s + 14.2·49-s − 2·53-s − 55-s + 8·59-s + 7.21·61-s + 4.60·65-s + 4·67-s − 14.4·71-s + 0.605·73-s − 4.60·77-s + 11.2·79-s − 10.6·83-s + 6.60·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.74·7-s + 0.301·11-s − 1.27·13-s − 1.60·17-s + 1.65·19-s + 0.200·25-s − 1.48·29-s − 1.65·31-s + 0.778·35-s − 0.527·37-s − 1.24·41-s + 0.517·43-s − 0.760·47-s + 2.03·49-s − 0.274·53-s − 0.134·55-s + 1.04·59-s + 0.923·61-s + 0.571·65-s + 0.488·67-s − 1.71·71-s + 0.0708·73-s − 0.524·77-s + 1.26·79-s − 1.16·83-s + 0.716·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4316592513\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4316592513\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 + 4.60T + 7T^{2} \) |
| 13 | \( 1 + 4.60T + 13T^{2} \) |
| 17 | \( 1 + 6.60T + 17T^{2} \) |
| 19 | \( 1 - 7.21T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 + 9.21T + 31T^{2} \) |
| 37 | \( 1 + 3.21T + 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 - 3.39T + 43T^{2} \) |
| 47 | \( 1 + 5.21T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 - 7.21T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 - 0.605T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−7.60017366118396307950429390639, −7.00891872598201563558799975826, −6.77102552405242496745151760111, −5.70158671667094594990176892892, −5.15385892152145313954897201307, −4.11232525833053061653749322358, −3.50025555899855534814454861781, −2.82983682538495676475230197901, −1.87555099932477660171282438902, −0.30564685641399426488259525472,
0.30564685641399426488259525472, 1.87555099932477660171282438902, 2.82983682538495676475230197901, 3.50025555899855534814454861781, 4.11232525833053061653749322358, 5.15385892152145313954897201307, 5.70158671667094594990176892892, 6.77102552405242496745151760111, 7.00891872598201563558799975826, 7.60017366118396307950429390639
