L(s) = 1 | − 2.54·2-s + 2.46·3-s + 4.48·4-s − 2.78·5-s − 6.26·6-s − 7-s − 6.31·8-s + 3.06·9-s + 7.09·10-s + 4.70·11-s + 11.0·12-s − 2.32·13-s + 2.54·14-s − 6.86·15-s + 7.11·16-s + 1.82·17-s − 7.80·18-s − 7.09·19-s − 12.4·20-s − 2.46·21-s − 11.9·22-s − 15.5·24-s + 2.77·25-s + 5.92·26-s + 0.159·27-s − 4.48·28-s − 9.98·29-s + ⋯ |
L(s) = 1 | − 1.80·2-s + 1.42·3-s + 2.24·4-s − 1.24·5-s − 2.55·6-s − 0.377·7-s − 2.23·8-s + 1.02·9-s + 2.24·10-s + 1.41·11-s + 3.18·12-s − 0.645·13-s + 0.680·14-s − 1.77·15-s + 1.77·16-s + 0.443·17-s − 1.83·18-s − 1.62·19-s − 2.79·20-s − 0.537·21-s − 2.55·22-s − 3.17·24-s + 0.555·25-s + 1.16·26-s + 0.0307·27-s − 0.846·28-s − 1.85·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3703 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3703 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8736057606\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8736057606\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + 2.54T + 2T^{2} \) |
| 3 | \( 1 - 2.46T + 3T^{2} \) |
| 5 | \( 1 + 2.78T + 5T^{2} \) |
| 11 | \( 1 - 4.70T + 11T^{2} \) |
| 13 | \( 1 + 2.32T + 13T^{2} \) |
| 17 | \( 1 - 1.82T + 17T^{2} \) |
| 19 | \( 1 + 7.09T + 19T^{2} \) |
| 29 | \( 1 + 9.98T + 29T^{2} \) |
| 31 | \( 1 - 3.53T + 31T^{2} \) |
| 37 | \( 1 - 0.166T + 37T^{2} \) |
| 41 | \( 1 - 7.25T + 41T^{2} \) |
| 43 | \( 1 - 9.57T + 43T^{2} \) |
| 47 | \( 1 - 4.66T + 47T^{2} \) |
| 53 | \( 1 - 0.961T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 + 0.954T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 - 4.59T + 71T^{2} \) |
| 73 | \( 1 + 7.59T + 73T^{2} \) |
| 79 | \( 1 + 5.73T + 79T^{2} \) |
| 83 | \( 1 - 5.57T + 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 - 1.68T + 97T^{2} \) |
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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
−8.509727261066999440054541000974, −8.031066384887242100688144319569, −7.38369405047526022893569940436, −6.93276109745443284662765564194, −5.98465912162994026267248295636, −4.13684167405699751017445883358, −3.75818639641609945364016092285, −2.65076253589591555869552374864, −1.93978242527522802039228711509, −0.64953836160440086840436403296,
0.64953836160440086840436403296, 1.93978242527522802039228711509, 2.65076253589591555869552374864, 3.75818639641609945364016092285, 4.13684167405699751017445883358, 5.98465912162994026267248295636, 6.93276109745443284662765564194, 7.38369405047526022893569940436, 8.031066384887242100688144319569, 8.509727261066999440054541000974