| L(s) = 1 | + 0.329·2-s − 1.89·4-s + 2.73·5-s − 7-s − 1.28·8-s + 0.902·10-s + 2.50·11-s + 1.48·13-s − 0.329·14-s + 3.35·16-s − 0.902·17-s + 2.50·19-s − 5.17·20-s + 0.825·22-s − 23-s + 2.48·25-s + 0.489·26-s + 1.89·28-s − 6.68·29-s + 1.09·31-s + 3.67·32-s − 0.297·34-s − 2.73·35-s + 9.91·37-s + 0.825·38-s − 3.51·40-s + 2.30·41-s + ⋯ |
| L(s) = 1 | + 0.233·2-s − 0.945·4-s + 1.22·5-s − 0.377·7-s − 0.453·8-s + 0.285·10-s + 0.755·11-s + 0.411·13-s − 0.0881·14-s + 0.839·16-s − 0.218·17-s + 0.574·19-s − 1.15·20-s + 0.176·22-s − 0.208·23-s + 0.497·25-s + 0.0960·26-s + 0.357·28-s − 1.24·29-s + 0.197·31-s + 0.649·32-s − 0.0510·34-s − 0.462·35-s + 1.62·37-s + 0.133·38-s − 0.555·40-s + 0.360·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.898284909\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.898284909\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 0.329T + 2T^{2} \) |
| 5 | \( 1 - 2.73T + 5T^{2} \) |
| 11 | \( 1 - 2.50T + 11T^{2} \) |
| 13 | \( 1 - 1.48T + 13T^{2} \) |
| 17 | \( 1 + 0.902T + 17T^{2} \) |
| 19 | \( 1 - 2.50T + 19T^{2} \) |
| 29 | \( 1 + 6.68T + 29T^{2} \) |
| 31 | \( 1 - 1.09T + 31T^{2} \) |
| 37 | \( 1 - 9.91T + 37T^{2} \) |
| 41 | \( 1 - 2.30T + 41T^{2} \) |
| 43 | \( 1 - 5.83T + 43T^{2} \) |
| 47 | \( 1 - 6.74T + 47T^{2} \) |
| 53 | \( 1 - 4.91T + 53T^{2} \) |
| 59 | \( 1 - 7.32T + 59T^{2} \) |
| 61 | \( 1 + 1.00T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 - 0.362T + 71T^{2} \) |
| 73 | \( 1 + 5.34T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 + 7.59T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 - 13.9T + 97T^{2} \) |
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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
−9.437469801611656023137237262824, −9.056799947983244638644670312182, −8.047440028888771842524548870581, −6.93992655687973082904644429005, −5.90454558344888556621945536742, −5.62669760755407085579022456633, −4.40135063764434433227127237481, −3.62196139763904584394000486757, −2.37848360070806315938870367081, −1.01410313391999095923333666846,
1.01410313391999095923333666846, 2.37848360070806315938870367081, 3.62196139763904584394000486757, 4.40135063764434433227127237481, 5.62669760755407085579022456633, 5.90454558344888556621945536742, 6.93992655687973082904644429005, 8.047440028888771842524548870581, 9.056799947983244638644670312182, 9.437469801611656023137237262824
