L(s) = 1 | − 3-s − 2.37·5-s + 7-s + 9-s + 4.37·11-s − 13-s + 2.37·15-s + 4.37·17-s + 2.37·19-s − 21-s − 6.37·23-s + 0.627·25-s − 27-s + 4.37·29-s − 4·31-s − 4.37·33-s − 2.37·35-s + 3.62·37-s + 39-s + 8·41-s + 1.62·43-s − 2.37·45-s − 6.74·47-s + 49-s − 4.37·51-s + 6·53-s − 10.3·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.06·5-s + 0.377·7-s + 0.333·9-s + 1.31·11-s − 0.277·13-s + 0.612·15-s + 1.06·17-s + 0.544·19-s − 0.218·21-s − 1.32·23-s + 0.125·25-s − 0.192·27-s + 0.811·29-s − 0.718·31-s − 0.761·33-s − 0.400·35-s + 0.596·37-s + 0.160·39-s + 1.24·41-s + 0.248·43-s − 0.353·45-s − 0.983·47-s + 0.142·49-s − 0.612·51-s + 0.824·53-s − 1.39·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.344599193\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.344599193\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 + 2.37T + 5T^{2} \) |
| 11 | \( 1 - 4.37T + 11T^{2} \) |
| 17 | \( 1 - 4.37T + 17T^{2} \) |
| 19 | \( 1 - 2.37T + 19T^{2} \) |
| 23 | \( 1 + 6.37T + 23T^{2} \) |
| 29 | \( 1 - 4.37T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 3.62T + 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 - 1.62T + 43T^{2} \) |
| 47 | \( 1 + 6.74T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 6.74T + 59T^{2} \) |
| 61 | \( 1 + 5.11T + 61T^{2} \) |
| 67 | \( 1 - 8.74T + 67T^{2} \) |
| 71 | \( 1 + 6.74T + 71T^{2} \) |
| 73 | \( 1 - 0.372T + 73T^{2} \) |
| 79 | \( 1 + 4.74T + 79T^{2} \) |
| 83 | \( 1 - 2T + 83T^{2} \) |
| 89 | \( 1 + 8.74T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−8.124601962254785369902042214984, −7.69665069133783332428996171488, −6.98752792048629997736321482333, −6.13419546947065341026689862272, −5.47994746802143115910409983209, −4.42613796915471136825504066282, −4.00899669940403824413288000440, −3.12088017152515090058115103166, −1.70858833268963562478390790873, −0.70137363351857694340745819859,
0.70137363351857694340745819859, 1.70858833268963562478390790873, 3.12088017152515090058115103166, 4.00899669940403824413288000440, 4.42613796915471136825504066282, 5.47994746802143115910409983209, 6.13419546947065341026689862272, 6.98752792048629997736321482333, 7.69665069133783332428996171488, 8.124601962254785369902042214984