L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s − 11-s + 5.46·13-s − 14-s + 16-s − 3.46·17-s + 3.26·19-s + 20-s + 22-s − 2.19·23-s + 25-s − 5.46·26-s + 28-s + 1.26·29-s + 2·31-s − 32-s + 3.46·34-s + 35-s − 2.73·37-s − 3.26·38-s − 40-s − 8.19·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s − 0.301·11-s + 1.51·13-s − 0.267·14-s + 0.250·16-s − 0.840·17-s + 0.749·19-s + 0.223·20-s + 0.213·22-s − 0.457·23-s + 0.200·25-s − 1.07·26-s + 0.188·28-s + 0.235·29-s + 0.359·31-s − 0.176·32-s + 0.594·34-s + 0.169·35-s − 0.449·37-s − 0.530·38-s − 0.158·40-s − 1.28·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.762456610\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.762456610\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 - 5.46T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 3.26T + 19T^{2} \) |
| 23 | \( 1 + 2.19T + 23T^{2} \) |
| 29 | \( 1 - 1.26T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 2.73T + 37T^{2} \) |
| 41 | \( 1 + 8.19T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 - 6.92T + 59T^{2} \) |
| 61 | \( 1 - 8.92T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 2.53T + 71T^{2} \) |
| 73 | \( 1 - 6.39T + 73T^{2} \) |
| 79 | \( 1 + 1.80T + 79T^{2} \) |
| 83 | \( 1 + 4.39T + 83T^{2} \) |
| 89 | \( 1 - 3.46T + 89T^{2} \) |
| 97 | \( 1 + 16.5T + 97T^{2} \) |
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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
−8.251938760688604696883783565364, −7.21839684304453786256179730672, −6.71119733089994755466345633731, −5.86688225846272254187912827572, −5.36326074032533073149393101568, −4.30578912311957278223543666010, −3.49208357857907319786904778050, −2.51212391152536451503767539070, −1.69057229522510183493212845516, −0.78387952088593746868922185064,
0.78387952088593746868922185064, 1.69057229522510183493212845516, 2.51212391152536451503767539070, 3.49208357857907319786904778050, 4.30578912311957278223543666010, 5.36326074032533073149393101568, 5.86688225846272254187912827572, 6.71119733089994755466345633731, 7.21839684304453786256179730672, 8.251938760688604696883783565364