L(s) = 1 | + 0.741·2-s − 1.45·4-s − 1.03·7-s − 2.55·8-s + 0.513·11-s − 3.54·13-s − 0.767·14-s + 1.00·16-s − 1.36·17-s + 0.894·19-s + 0.380·22-s + 5.45·23-s − 2.62·26-s + 1.50·28-s + 9.65·29-s + 10.4·31-s + 5.86·32-s − 1.01·34-s + 2.19·37-s + 0.663·38-s − 3.12·41-s − 10.2·43-s − 0.745·44-s + 4.04·46-s + 7.65·47-s − 5.92·49-s + 5.13·52-s + ⋯ |
L(s) = 1 | + 0.524·2-s − 0.725·4-s − 0.391·7-s − 0.904·8-s + 0.154·11-s − 0.982·13-s − 0.205·14-s + 0.251·16-s − 0.331·17-s + 0.205·19-s + 0.0812·22-s + 1.13·23-s − 0.515·26-s + 0.283·28-s + 1.79·29-s + 1.87·31-s + 1.03·32-s − 0.173·34-s + 0.360·37-s + 0.107·38-s − 0.487·41-s − 1.56·43-s − 0.112·44-s + 0.596·46-s + 1.11·47-s − 0.846·49-s + 0.712·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 0.741T + 2T^{2} \) |
| 7 | \( 1 + 1.03T + 7T^{2} \) |
| 11 | \( 1 - 0.513T + 11T^{2} \) |
| 13 | \( 1 + 3.54T + 13T^{2} \) |
| 17 | \( 1 + 1.36T + 17T^{2} \) |
| 19 | \( 1 - 0.894T + 19T^{2} \) |
| 23 | \( 1 - 5.45T + 23T^{2} \) |
| 29 | \( 1 - 9.65T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 - 2.19T + 37T^{2} \) |
| 41 | \( 1 + 3.12T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 - 7.65T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 - 7.85T + 61T^{2} \) |
| 67 | \( 1 + 8.80T + 67T^{2} \) |
| 71 | \( 1 + 5.00T + 71T^{2} \) |
| 73 | \( 1 + 5.82T + 73T^{2} \) |
| 79 | \( 1 + 6.74T + 79T^{2} \) |
| 83 | \( 1 + 7.99T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 - 7.27T + 97T^{2} \) |
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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
−7.87556995569003887558723294286, −6.82614618777547837634111232430, −6.40573203796976426632697783408, −5.45984377492927637238428807193, −4.67107938552227595223584422252, −4.40528608032660642504963803023, −3.08167016805999526600974950944, −2.82647771875711633003213684109, −1.22354910764416862564520334922, 0,
1.22354910764416862564520334922, 2.82647771875711633003213684109, 3.08167016805999526600974950944, 4.40528608032660642504963803023, 4.67107938552227595223584422252, 5.45984377492927637238428807193, 6.40573203796976426632697783408, 6.82614618777547837634111232430, 7.87556995569003887558723294286