| L(s) = 1 | + 5-s − 0.334·7-s − 2.12·11-s − 2.25·13-s − 6.93·17-s + 4.12·19-s + 23-s + 25-s + 1.92·29-s + 0.440·31-s − 0.334·35-s + 6.04·37-s + 10.2·41-s − 5.50·43-s + 4.65·47-s − 6.88·49-s + 4.40·53-s − 2.12·55-s + 10.8·59-s + 13.7·61-s − 2.25·65-s + 6.87·67-s − 5.57·71-s + 6.25·73-s + 0.709·77-s + 16.7·79-s − 10.2·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.126·7-s − 0.639·11-s − 0.625·13-s − 1.68·17-s + 0.945·19-s + 0.208·23-s + 0.200·25-s + 0.356·29-s + 0.0791·31-s − 0.0565·35-s + 0.993·37-s + 1.60·41-s − 0.838·43-s + 0.679·47-s − 0.984·49-s + 0.604·53-s − 0.286·55-s + 1.41·59-s + 1.76·61-s − 0.279·65-s + 0.839·67-s − 0.662·71-s + 0.732·73-s + 0.0808·77-s + 1.87·79-s − 1.12·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.736222897\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.736222897\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 0.334T + 7T^{2} \) |
| 11 | \( 1 + 2.12T + 11T^{2} \) |
| 13 | \( 1 + 2.25T + 13T^{2} \) |
| 17 | \( 1 + 6.93T + 17T^{2} \) |
| 19 | \( 1 - 4.12T + 19T^{2} \) |
| 29 | \( 1 - 1.92T + 29T^{2} \) |
| 31 | \( 1 - 0.440T + 31T^{2} \) |
| 37 | \( 1 - 6.04T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 + 5.50T + 43T^{2} \) |
| 47 | \( 1 - 4.65T + 47T^{2} \) |
| 53 | \( 1 - 4.40T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 - 6.87T + 67T^{2} \) |
| 71 | \( 1 + 5.57T + 71T^{2} \) |
| 73 | \( 1 - 6.25T + 73T^{2} \) |
| 79 | \( 1 - 16.7T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 + 1.33T + 89T^{2} \) |
| 97 | \( 1 - 18.4T + 97T^{2} \) |
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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
−8.444502610459773431716253406883, −7.65397636953391449947912157801, −6.93218720182651645121107895922, −6.27553197654632799128240010510, −5.36449246951885797915851095903, −4.78535092781026978522172288015, −3.87061606167528162462517371709, −2.70724901188590614156795148511, −2.18577620187739833051955038335, −0.73145949031201432934140032293,
0.73145949031201432934140032293, 2.18577620187739833051955038335, 2.70724901188590614156795148511, 3.87061606167528162462517371709, 4.78535092781026978522172288015, 5.36449246951885797915851095903, 6.27553197654632799128240010510, 6.93218720182651645121107895922, 7.65397636953391449947912157801, 8.444502610459773431716253406883
