| L(s) = 1 | + 2.09·2-s + 7.20·3-s − 3.60·4-s + 16.1·5-s + 15.1·6-s − 24.3·8-s + 24.9·9-s + 33.8·10-s − 54.7·11-s − 25.9·12-s + 16.1·13-s + 116.·15-s − 22.1·16-s − 14.2·17-s + 52.2·18-s − 110.·19-s − 58.1·20-s − 114.·22-s − 113.·23-s − 175.·24-s + 135.·25-s + 33.7·26-s − 14.8·27-s − 125.·29-s + 243.·30-s − 121.·31-s + 148.·32-s + ⋯ |
| L(s) = 1 | + 0.741·2-s + 1.38·3-s − 0.450·4-s + 1.44·5-s + 1.02·6-s − 1.07·8-s + 0.923·9-s + 1.06·10-s − 1.50·11-s − 0.624·12-s + 0.343·13-s + 2.00·15-s − 0.346·16-s − 0.202·17-s + 0.684·18-s − 1.33·19-s − 0.649·20-s − 1.11·22-s − 1.03·23-s − 1.49·24-s + 1.08·25-s + 0.254·26-s − 0.105·27-s − 0.801·29-s + 1.48·30-s − 0.701·31-s + 0.818·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 + 41T \) |
| good | 2 | \( 1 - 2.09T + 8T^{2} \) |
| 3 | \( 1 - 7.20T + 27T^{2} \) |
| 5 | \( 1 - 16.1T + 125T^{2} \) |
| 11 | \( 1 + 54.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 16.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 14.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 110.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 113.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 125.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 121.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 259.T + 5.06e4T^{2} \) |
| 43 | \( 1 + 436.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 185.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 480.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 565.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 454.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 584.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 690.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 4.41T + 3.89e5T^{2} \) |
| 79 | \( 1 - 792.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 393.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 936.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.31T + 9.12e5T^{2} \) |
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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
−8.383683073518298803429201129808, −7.978551878191041740599624550942, −6.64222852217304264094877222464, −5.82330962050666223553823644160, −5.20367251913471580740332910907, −4.20719908772128603799723712610, −3.31697054927938069173089320792, −2.42997215223438116678266338753, −1.90837797248489630912140691630, 0,
1.90837797248489630912140691630, 2.42997215223438116678266338753, 3.31697054927938069173089320792, 4.20719908772128603799723712610, 5.20367251913471580740332910907, 5.82330962050666223553823644160, 6.64222852217304264094877222464, 7.978551878191041740599624550942, 8.383683073518298803429201129808
