| L(s) = 1 | + 2.76·2-s − 3.30·3-s + 5.61·4-s − 9.11·6-s + 4.87·7-s + 9.98·8-s + 7.90·9-s + 1.41·11-s − 18.5·12-s + 0.466·13-s + 13.4·14-s + 16.3·16-s + 0.431·17-s + 21.8·18-s + 3.96·19-s − 16.0·21-s + 3.91·22-s + 2.67·23-s − 32.9·24-s + 1.28·26-s − 16.1·27-s + 27.3·28-s + 6.57·29-s + 2.31·31-s + 25.0·32-s − 4.68·33-s + 1.19·34-s + ⋯ |
| L(s) = 1 | + 1.95·2-s − 1.90·3-s + 2.80·4-s − 3.72·6-s + 1.84·7-s + 3.53·8-s + 2.63·9-s + 0.427·11-s − 5.35·12-s + 0.129·13-s + 3.59·14-s + 4.08·16-s + 0.104·17-s + 5.14·18-s + 0.909·19-s − 3.50·21-s + 0.834·22-s + 0.558·23-s − 6.73·24-s + 0.252·26-s − 3.11·27-s + 5.17·28-s + 1.22·29-s + 0.414·31-s + 4.43·32-s − 0.815·33-s + 0.204·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.075074413\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.075074413\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 - 2.76T + 2T^{2} \) |
| 3 | \( 1 + 3.30T + 3T^{2} \) |
| 7 | \( 1 - 4.87T + 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 - 0.466T + 13T^{2} \) |
| 17 | \( 1 - 0.431T + 17T^{2} \) |
| 19 | \( 1 - 3.96T + 19T^{2} \) |
| 23 | \( 1 - 2.67T + 23T^{2} \) |
| 29 | \( 1 - 6.57T + 29T^{2} \) |
| 31 | \( 1 - 2.31T + 31T^{2} \) |
| 37 | \( 1 + 9.54T + 37T^{2} \) |
| 41 | \( 1 + 2.14T + 41T^{2} \) |
| 43 | \( 1 + 9.72T + 43T^{2} \) |
| 47 | \( 1 - 0.849T + 47T^{2} \) |
| 53 | \( 1 + 0.687T + 53T^{2} \) |
| 59 | \( 1 + 8.86T + 59T^{2} \) |
| 61 | \( 1 + 3.50T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 - 4.14T + 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 + 4.40T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 + 6.36T + 89T^{2} \) |
| 97 | \( 1 - 6.27T + 97T^{2} \) |
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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
−7.49916020508353356272344468175, −7.01701785911096466298838893867, −6.36988292301031265784073166138, −5.59402024137716494065503287159, −5.17583541567958776597916320322, −4.65572829328184558257513906557, −4.20844618503211554443697211215, −3.09040752296563925828740226338, −1.63372445259429234666345323949, −1.30288304283677562334955457901,
1.30288304283677562334955457901, 1.63372445259429234666345323949, 3.09040752296563925828740226338, 4.20844618503211554443697211215, 4.65572829328184558257513906557, 5.17583541567958776597916320322, 5.59402024137716494065503287159, 6.36988292301031265784073166138, 7.01701785911096466298838893867, 7.49916020508353356272344468175
