| L(s) = 1 | − 0.585·5-s − 0.828·11-s − 1.41·13-s − 2.24·17-s − 6.82·19-s + 4.82·23-s − 4.65·25-s − 8.48·29-s − 5.17·31-s + 1.65·37-s + 0.585·41-s + 8·43-s + 6.82·47-s + 13.3·53-s + 0.485·55-s + 5.17·59-s + 13.8·61-s + 0.828·65-s + 8·67-s − 0.828·71-s − 11.0·73-s + 2.34·79-s + 15.3·83-s + 1.31·85-s + 10.7·89-s + 4·95-s − 7.75·97-s + ⋯ |
| L(s) = 1 | − 0.261·5-s − 0.249·11-s − 0.392·13-s − 0.543·17-s − 1.56·19-s + 1.00·23-s − 0.931·25-s − 1.57·29-s − 0.928·31-s + 0.272·37-s + 0.0914·41-s + 1.21·43-s + 0.996·47-s + 1.82·53-s + 0.0654·55-s + 0.673·59-s + 1.77·61-s + 0.102·65-s + 0.977·67-s − 0.0983·71-s − 1.29·73-s + 0.263·79-s + 1.68·83-s + 0.142·85-s + 1.13·89-s + 0.410·95-s − 0.787·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.265442711\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.265442711\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 + 0.585T + 5T^{2} \) |
| 11 | \( 1 + 0.828T + 11T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 + 2.24T + 17T^{2} \) |
| 19 | \( 1 + 6.82T + 19T^{2} \) |
| 23 | \( 1 - 4.82T + 23T^{2} \) |
| 29 | \( 1 + 8.48T + 29T^{2} \) |
| 31 | \( 1 + 5.17T + 31T^{2} \) |
| 37 | \( 1 - 1.65T + 37T^{2} \) |
| 41 | \( 1 - 0.585T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 6.82T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 - 5.17T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 0.828T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 - 2.34T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 + 7.75T + 97T^{2} \) |
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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
−7.84559382173864157893131293476, −7.27535810676448797876926330767, −6.65237302822365979181137794291, −5.74235152952742374041253616764, −5.21169154175944496043081488782, −4.13978158786338060007048669129, −3.82652219361121838120568632629, −2.53774985939799479818192528144, −2.00266703295827614311349636913, −0.54688236611422705431268569970,
0.54688236611422705431268569970, 2.00266703295827614311349636913, 2.53774985939799479818192528144, 3.82652219361121838120568632629, 4.13978158786338060007048669129, 5.21169154175944496043081488782, 5.74235152952742374041253616764, 6.65237302822365979181137794291, 7.27535810676448797876926330767, 7.84559382173864157893131293476
