L(s) = 1 | + 3.61·5-s + 3.03·7-s + 1.68·11-s − 4.92·13-s + 1.55·17-s + 5.19·19-s + 2.10·23-s + 8.06·25-s − 2.89·29-s + 3.08·31-s + 10.9·35-s − 0.374·37-s − 11.7·41-s + 3.97·43-s + 6.21·47-s + 2.19·49-s + 6.03·53-s + 6.09·55-s − 59-s + 4.67·61-s − 17.8·65-s − 5.13·67-s + 2.17·71-s − 0.761·73-s + 5.11·77-s − 6.64·79-s − 3.43·83-s + ⋯ |
L(s) = 1 | + 1.61·5-s + 1.14·7-s + 0.508·11-s − 1.36·13-s + 0.376·17-s + 1.19·19-s + 0.438·23-s + 1.61·25-s − 0.537·29-s + 0.554·31-s + 1.85·35-s − 0.0615·37-s − 1.84·41-s + 0.605·43-s + 0.905·47-s + 0.313·49-s + 0.829·53-s + 0.821·55-s − 0.130·59-s + 0.598·61-s − 2.20·65-s − 0.627·67-s + 0.258·71-s − 0.0891·73-s + 0.582·77-s − 0.747·79-s − 0.377·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.676723624\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.676723624\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 59 | \( 1 + T \) |
good | 5 | \( 1 - 3.61T + 5T^{2} \) |
| 7 | \( 1 - 3.03T + 7T^{2} \) |
| 11 | \( 1 - 1.68T + 11T^{2} \) |
| 13 | \( 1 + 4.92T + 13T^{2} \) |
| 17 | \( 1 - 1.55T + 17T^{2} \) |
| 19 | \( 1 - 5.19T + 19T^{2} \) |
| 23 | \( 1 - 2.10T + 23T^{2} \) |
| 29 | \( 1 + 2.89T + 29T^{2} \) |
| 31 | \( 1 - 3.08T + 31T^{2} \) |
| 37 | \( 1 + 0.374T + 37T^{2} \) |
| 41 | \( 1 + 11.7T + 41T^{2} \) |
| 43 | \( 1 - 3.97T + 43T^{2} \) |
| 47 | \( 1 - 6.21T + 47T^{2} \) |
| 53 | \( 1 - 6.03T + 53T^{2} \) |
| 61 | \( 1 - 4.67T + 61T^{2} \) |
| 67 | \( 1 + 5.13T + 67T^{2} \) |
| 71 | \( 1 - 2.17T + 71T^{2} \) |
| 73 | \( 1 + 0.761T + 73T^{2} \) |
| 79 | \( 1 + 6.64T + 79T^{2} \) |
| 83 | \( 1 + 3.43T + 83T^{2} \) |
| 89 | \( 1 - 7.64T + 89T^{2} \) |
| 97 | \( 1 - 18.7T + 97T^{2} \) |
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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
−7.60511586078262945242847994347, −7.19559651279589281580852538071, −6.34706759289083586599122499913, −5.53592436498234286566159450230, −5.14806964059307002509565089085, −4.54248586502639676023593659251, −3.33970831080009578579359353091, −2.43282117564923019880617367777, −1.79601906863582295401029712412, −1.00716093954140169089157765662,
1.00716093954140169089157765662, 1.79601906863582295401029712412, 2.43282117564923019880617367777, 3.33970831080009578579359353091, 4.54248586502639676023593659251, 5.14806964059307002509565089085, 5.53592436498234286566159450230, 6.34706759289083586599122499913, 7.19559651279589281580852538071, 7.60511586078262945242847994347