| L(s) = 1 | − 0.577·3-s + 2.10·5-s − 1.73·7-s − 2.66·9-s − 6.36·11-s − 2.18·13-s − 1.21·15-s − 0.631·17-s + 7.04·19-s + 1.00·21-s − 6.00·23-s − 0.563·25-s + 3.27·27-s + 7.71·29-s − 0.467·31-s + 3.67·33-s − 3.65·35-s + 2.99·37-s + 1.26·39-s − 8.67·41-s + 11.0·43-s − 5.61·45-s − 4.33·47-s − 3.98·49-s + 0.364·51-s − 8.09·53-s − 13.4·55-s + ⋯ |
| L(s) = 1 | − 0.333·3-s + 0.942·5-s − 0.655·7-s − 0.888·9-s − 1.91·11-s − 0.605·13-s − 0.314·15-s − 0.153·17-s + 1.61·19-s + 0.218·21-s − 1.25·23-s − 0.112·25-s + 0.630·27-s + 1.43·29-s − 0.0840·31-s + 0.640·33-s − 0.617·35-s + 0.492·37-s + 0.202·39-s − 1.35·41-s + 1.68·43-s − 0.837·45-s − 0.631·47-s − 0.569·49-s + 0.0510·51-s − 1.11·53-s − 1.80·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9665690683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9665690683\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 + 0.577T + 3T^{2} \) |
| 5 | \( 1 - 2.10T + 5T^{2} \) |
| 7 | \( 1 + 1.73T + 7T^{2} \) |
| 11 | \( 1 + 6.36T + 11T^{2} \) |
| 13 | \( 1 + 2.18T + 13T^{2} \) |
| 17 | \( 1 + 0.631T + 17T^{2} \) |
| 19 | \( 1 - 7.04T + 19T^{2} \) |
| 23 | \( 1 + 6.00T + 23T^{2} \) |
| 29 | \( 1 - 7.71T + 29T^{2} \) |
| 31 | \( 1 + 0.467T + 31T^{2} \) |
| 37 | \( 1 - 2.99T + 37T^{2} \) |
| 41 | \( 1 + 8.67T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 + 4.33T + 47T^{2} \) |
| 53 | \( 1 + 8.09T + 53T^{2} \) |
| 59 | \( 1 + 7.20T + 59T^{2} \) |
| 61 | \( 1 - 5.93T + 61T^{2} \) |
| 67 | \( 1 - 2.21T + 67T^{2} \) |
| 71 | \( 1 - 5.83T + 71T^{2} \) |
| 73 | \( 1 + 9.72T + 73T^{2} \) |
| 79 | \( 1 + 1.62T + 79T^{2} \) |
| 83 | \( 1 + 1.72T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 - 7.97T + 97T^{2} \) |
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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
−7.940819349770710647471305239051, −7.09201255383377280574516612948, −6.26617456528016789344292344477, −5.67120627030520482010193434365, −5.28480259603485391416618354719, −4.52617321811978364381604711254, −3.09774947954473774611068636235, −2.81782031957357913731203874704, −1.91445769579419568696128603948, −0.46363760562161839635562353436,
0.46363760562161839635562353436, 1.91445769579419568696128603948, 2.81782031957357913731203874704, 3.09774947954473774611068636235, 4.52617321811978364381604711254, 5.28480259603485391416618354719, 5.67120627030520482010193434365, 6.26617456528016789344292344477, 7.09201255383377280574516612948, 7.940819349770710647471305239051
