L(s) = 1 | − 2.50·3-s + 2.93·5-s − 0.515·7-s + 3.27·9-s + 4.74·11-s + 0.532·13-s − 7.36·15-s + 17-s + 4.25·19-s + 1.29·21-s + 8.33·23-s + 3.64·25-s − 0.680·27-s − 2.15·29-s + 0.646·31-s − 11.8·33-s − 1.51·35-s − 5.29·37-s − 1.33·39-s − 5.05·41-s + 7.82·43-s + 9.61·45-s + 7.28·47-s − 6.73·49-s − 2.50·51-s + 3.75·53-s + 13.9·55-s + ⋯ |
L(s) = 1 | − 1.44·3-s + 1.31·5-s − 0.194·7-s + 1.09·9-s + 1.43·11-s + 0.147·13-s − 1.90·15-s + 0.242·17-s + 0.975·19-s + 0.281·21-s + 1.73·23-s + 0.728·25-s − 0.131·27-s − 0.400·29-s + 0.116·31-s − 2.06·33-s − 0.256·35-s − 0.870·37-s − 0.213·39-s − 0.789·41-s + 1.19·43-s + 1.43·45-s + 1.06·47-s − 0.962·49-s − 0.350·51-s + 0.515·53-s + 1.88·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.764096547\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.764096547\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
good | 3 | \( 1 + 2.50T + 3T^{2} \) |
| 5 | \( 1 - 2.93T + 5T^{2} \) |
| 7 | \( 1 + 0.515T + 7T^{2} \) |
| 11 | \( 1 - 4.74T + 11T^{2} \) |
| 13 | \( 1 - 0.532T + 13T^{2} \) |
| 19 | \( 1 - 4.25T + 19T^{2} \) |
| 23 | \( 1 - 8.33T + 23T^{2} \) |
| 29 | \( 1 + 2.15T + 29T^{2} \) |
| 31 | \( 1 - 0.646T + 31T^{2} \) |
| 37 | \( 1 + 5.29T + 37T^{2} \) |
| 41 | \( 1 + 5.05T + 41T^{2} \) |
| 43 | \( 1 - 7.82T + 43T^{2} \) |
| 47 | \( 1 - 7.28T + 47T^{2} \) |
| 53 | \( 1 - 3.75T + 53T^{2} \) |
| 61 | \( 1 - 8.88T + 61T^{2} \) |
| 67 | \( 1 + 4.98T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 + 0.881T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 - 1.63T + 83T^{2} \) |
| 89 | \( 1 + 8.25T + 89T^{2} \) |
| 97 | \( 1 - 5.34T + 97T^{2} \) |
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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
−8.740952649713837416744050063923, −7.31479075716033165878259089634, −6.75388634561921720109331035223, −6.17872495274548029048924397455, −5.50256283162977469399114658910, −5.06684497418618647405515212494, −3.99587756676030284544670979076, −2.92992343573165932560870590795, −1.59164591217644595215340025981, −0.908473046826744783819011805578,
0.908473046826744783819011805578, 1.59164591217644595215340025981, 2.92992343573165932560870590795, 3.99587756676030284544670979076, 5.06684497418618647405515212494, 5.50256283162977469399114658910, 6.17872495274548029048924397455, 6.75388634561921720109331035223, 7.31479075716033165878259089634, 8.740952649713837416744050063923