| L(s) = 1 | − 3-s − 3.41·5-s + 9-s + 0.828·11-s + 4.24·13-s + 3.41·15-s + 7.41·17-s + 6.82·19-s + 4.82·23-s + 6.65·25-s − 27-s − 2.82·29-s + 2.82·31-s − 0.828·33-s + 1.65·37-s − 4.24·39-s + 10.2·41-s − 11.3·43-s − 3.41·45-s + 4.48·47-s − 7.41·51-s + 2·53-s − 2.82·55-s − 6.82·57-s + 8.48·59-s − 11.0·61-s − 14.4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.52·5-s + 0.333·9-s + 0.249·11-s + 1.17·13-s + 0.881·15-s + 1.79·17-s + 1.56·19-s + 1.00·23-s + 1.33·25-s − 0.192·27-s − 0.525·29-s + 0.508·31-s − 0.144·33-s + 0.272·37-s − 0.679·39-s + 1.59·41-s − 1.72·43-s − 0.508·45-s + 0.654·47-s − 1.03·51-s + 0.274·53-s − 0.381·55-s − 0.904·57-s + 1.10·59-s − 1.41·61-s − 1.79·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.635540517\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.635540517\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 + 3.41T + 5T^{2} \) |
| 11 | \( 1 - 0.828T + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 - 7.41T + 17T^{2} \) |
| 19 | \( 1 - 6.82T + 19T^{2} \) |
| 23 | \( 1 - 4.82T + 23T^{2} \) |
| 29 | \( 1 + 2.82T + 29T^{2} \) |
| 31 | \( 1 - 2.82T + 31T^{2} \) |
| 37 | \( 1 - 1.65T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 - 4.48T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 8.48T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 - 7.75T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 5.75T + 89T^{2} \) |
| 97 | \( 1 + 0.242T + 97T^{2} \) |
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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
−7.71212231384432633819634952408, −7.14541147584709354759666860321, −6.38589310716060449822510360685, −5.52150892788813761471739189031, −5.05691908711372113357746931652, −4.03156196042972912551796396345, −3.57833832579480186289969532436, −2.94591525656567255632870769662, −1.25002761239093468326579984739, −0.76600126451800262411021570334,
0.76600126451800262411021570334, 1.25002761239093468326579984739, 2.94591525656567255632870769662, 3.57833832579480186289969532436, 4.03156196042972912551796396345, 5.05691908711372113357746931652, 5.52150892788813761471739189031, 6.38589310716060449822510360685, 7.14541147584709354759666860321, 7.71212231384432633819634952408
