| L(s) = 1 | − 2-s − 1.53·3-s + 4-s + 1.53·6-s − 0.347·7-s − 8-s + 1.34·9-s − 1.53·12-s + 1.87·13-s + 0.347·14-s + 16-s + 1.87·17-s − 1.34·18-s + 19-s + 0.532·21-s − 1.53·23-s + 1.53·24-s − 1.87·26-s − 0.532·27-s − 0.347·28-s − 1.87·29-s − 32-s − 1.87·34-s + 1.34·36-s + 37-s − 38-s − 2.87·39-s + ⋯ |
| L(s) = 1 | − 2-s − 1.53·3-s + 4-s + 1.53·6-s − 0.347·7-s − 8-s + 1.34·9-s − 1.53·12-s + 1.87·13-s + 0.347·14-s + 16-s + 1.87·17-s − 1.34·18-s + 19-s + 0.532·21-s − 1.53·23-s + 1.53·24-s − 1.87·26-s − 0.532·27-s − 0.347·28-s − 1.87·29-s − 32-s − 1.87·34-s + 1.34·36-s + 37-s − 38-s − 2.87·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5163551829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5163551829\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 1.53T + T^{2} \) |
| 7 | \( 1 + 0.347T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 1.87T + T^{2} \) |
| 17 | \( 1 - 1.87T + T^{2} \) |
| 23 | \( 1 + 1.53T + T^{2} \) |
| 29 | \( 1 + 1.87T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( 1 + 0.347T + T^{2} \) |
| 59 | \( 1 - 1.53T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 0.347T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.53T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.687242061115433114470497930237, −7.82044001749861364658649284334, −7.30571381588433022207473619224, −6.26773971781871546862797126903, −5.85353005407795758914816963419, −5.46406665589122244044972606026, −3.95353979828198557800762292202, −3.24307084052755935144829394953, −1.64419708545946417941928721076, −0.821521948736724844879885488022,
0.821521948736724844879885488022, 1.64419708545946417941928721076, 3.24307084052755935144829394953, 3.95353979828198557800762292202, 5.46406665589122244044972606026, 5.85353005407795758914816963419, 6.26773971781871546862797126903, 7.30571381588433022207473619224, 7.82044001749861364658649284334, 8.687242061115433114470497930237
