L(s) = 1 | − 2-s + 4-s − 5-s + 4.54·7-s − 8-s + 10-s + 2.54·11-s − 1.05·13-s − 4.54·14-s + 16-s + 4.39·17-s + 7.60·19-s − 20-s − 2.54·22-s + 1.20·23-s + 25-s + 1.05·26-s + 4.54·28-s − 6.10·29-s + 31-s − 32-s − 4.39·34-s − 4.54·35-s + 7.73·37-s − 7.60·38-s + 40-s − 6.10·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s + 1.71·7-s − 0.353·8-s + 0.316·10-s + 0.768·11-s − 0.291·13-s − 1.21·14-s + 0.250·16-s + 1.06·17-s + 1.74·19-s − 0.223·20-s − 0.543·22-s + 0.251·23-s + 0.200·25-s + 0.206·26-s + 0.859·28-s − 1.13·29-s + 0.179·31-s − 0.176·32-s − 0.753·34-s − 0.769·35-s + 1.27·37-s − 1.23·38-s + 0.158·40-s − 0.953·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.722632232\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.722632232\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 - T \) |
good | 7 | \( 1 - 4.54T + 7T^{2} \) |
| 11 | \( 1 - 2.54T + 11T^{2} \) |
| 13 | \( 1 + 1.05T + 13T^{2} \) |
| 17 | \( 1 - 4.39T + 17T^{2} \) |
| 19 | \( 1 - 7.60T + 19T^{2} \) |
| 23 | \( 1 - 1.20T + 23T^{2} \) |
| 29 | \( 1 + 6.10T + 29T^{2} \) |
| 37 | \( 1 - 7.73T + 37T^{2} \) |
| 41 | \( 1 + 6.10T + 41T^{2} \) |
| 43 | \( 1 + 6.28T + 43T^{2} \) |
| 47 | \( 1 + 2.68T + 47T^{2} \) |
| 53 | \( 1 - 0.502T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 + 9.80T + 61T^{2} \) |
| 67 | \( 1 - 7.46T + 67T^{2} \) |
| 71 | \( 1 - 8.65T + 71T^{2} \) |
| 73 | \( 1 + 8.70T + 73T^{2} \) |
| 79 | \( 1 + 0.918T + 79T^{2} \) |
| 83 | \( 1 + 2.41T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 + 9.20T + 97T^{2} \) |
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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
−8.704964740778554428564323147947, −7.947477590082471467212639198556, −7.59240548247915009182331370854, −6.83596306569965981690275784515, −5.59214028612187561373555415873, −5.04532554680886401281552173168, −4.01152825054695068992193717052, −3.04899269269836024182274431269, −1.72780972964094610765890792840, −1.00507496397471596218587421216,
1.00507496397471596218587421216, 1.72780972964094610765890792840, 3.04899269269836024182274431269, 4.01152825054695068992193717052, 5.04532554680886401281552173168, 5.59214028612187561373555415873, 6.83596306569965981690275784515, 7.59240548247915009182331370854, 7.947477590082471467212639198556, 8.704964740778554428564323147947