| L(s) = 1 | + 1.10·2-s − 0.777·4-s − 2.00·5-s + 3.50·7-s − 3.07·8-s − 2.22·10-s − 1.87·11-s + 5.47·13-s + 3.87·14-s − 1.84·16-s + 0.341·17-s + 2.18·19-s + 1.56·20-s − 2.07·22-s + 1.78·23-s − 0.966·25-s + 6.05·26-s − 2.72·28-s + 4.30·29-s + 4.49·31-s + 4.10·32-s + 0.377·34-s − 7.04·35-s + 3.77·37-s + 2.42·38-s + 6.16·40-s + 9.11·41-s + ⋯ |
| L(s) = 1 | + 0.781·2-s − 0.388·4-s − 0.898·5-s + 1.32·7-s − 1.08·8-s − 0.702·10-s − 0.565·11-s + 1.51·13-s + 1.03·14-s − 0.460·16-s + 0.0828·17-s + 0.502·19-s + 0.349·20-s − 0.442·22-s + 0.373·23-s − 0.193·25-s + 1.18·26-s − 0.515·28-s + 0.800·29-s + 0.806·31-s + 0.725·32-s + 0.0647·34-s − 1.19·35-s + 0.620·37-s + 0.392·38-s + 0.975·40-s + 1.42·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.030764151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.030764151\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 - 1.10T + 2T^{2} \) |
| 5 | \( 1 + 2.00T + 5T^{2} \) |
| 7 | \( 1 - 3.50T + 7T^{2} \) |
| 11 | \( 1 + 1.87T + 11T^{2} \) |
| 13 | \( 1 - 5.47T + 13T^{2} \) |
| 17 | \( 1 - 0.341T + 17T^{2} \) |
| 19 | \( 1 - 2.18T + 19T^{2} \) |
| 23 | \( 1 - 1.78T + 23T^{2} \) |
| 29 | \( 1 - 4.30T + 29T^{2} \) |
| 31 | \( 1 - 4.49T + 31T^{2} \) |
| 37 | \( 1 - 3.77T + 37T^{2} \) |
| 41 | \( 1 - 9.11T + 41T^{2} \) |
| 47 | \( 1 - 7.71T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 - 0.222T + 61T^{2} \) |
| 67 | \( 1 - 4.84T + 67T^{2} \) |
| 71 | \( 1 + 3.50T + 71T^{2} \) |
| 73 | \( 1 - 3.77T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 + 9.00T + 83T^{2} \) |
| 89 | \( 1 + 6.20T + 89T^{2} \) |
| 97 | \( 1 - 8.48T + 97T^{2} \) |
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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
−9.785550470743822681941650514602, −8.636125778711541670548009813978, −8.252884301792617197948308563446, −7.46105036978993247888854304811, −6.16373835871444128597436198956, −5.35545212768260701787744738174, −4.48315908851304280816551767973, −3.89975758235534587377911450926, −2.79493568600836844762982244729, −1.02139349850034862125089409901,
1.02139349850034862125089409901, 2.79493568600836844762982244729, 3.89975758235534587377911450926, 4.48315908851304280816551767973, 5.35545212768260701787744738174, 6.16373835871444128597436198956, 7.46105036978993247888854304811, 8.252884301792617197948308563446, 8.636125778711541670548009813978, 9.785550470743822681941650514602
