| L(s) = 1 | + 6·2-s + 14·4-s + 6·5-s + 14·7-s + 36·10-s + 48·11-s + 52·13-s + 84·14-s − 84·16-s + 30·17-s + 64·19-s + 84·20-s + 288·22-s + 60·23-s − 175·25-s + 312·26-s + 196·28-s + 360·29-s − 140·31-s − 216·32-s + 180·34-s + 84·35-s − 230·37-s + 384·38-s + 234·41-s − 938·43-s + 672·44-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 7/4·4-s + 0.536·5-s + 0.755·7-s + 1.13·10-s + 1.31·11-s + 1.10·13-s + 1.60·14-s − 1.31·16-s + 0.428·17-s + 0.772·19-s + 0.939·20-s + 2.79·22-s + 0.543·23-s − 7/5·25-s + 2.35·26-s + 1.32·28-s + 2.30·29-s − 0.811·31-s − 1.19·32-s + 0.907·34-s + 0.405·35-s − 1.02·37-s + 1.63·38-s + 0.891·41-s − 3.32·43-s + 2.30·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35721 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35721 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(9.376723892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.376723892\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - 3 p T + 11 p T^{2} - 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 6 T + 211 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 48 T + 2470 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 p T + 3342 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 30 T + 7699 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 64 T + 3942 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 60 T + 494 p T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 360 T + 77290 T^{2} - 360 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 140 T + 48930 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 230 T + 98979 T^{2} + 230 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 234 T + 26683 T^{2} - 234 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 938 T + 378543 T^{2} + 938 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 618 T + 210199 T^{2} - 618 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 420 T + 64606 T^{2} + 420 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 282 T + 411439 T^{2} - 282 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 32 T + 205386 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 544 T + 535542 T^{2} - 544 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 504 T + 736126 T^{2} + 504 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 764 T + 653958 T^{2} + 764 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 238 T + 125439 T^{2} - 238 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 522 T + 651823 T^{2} + 522 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 708 T + 1163542 T^{2} - 708 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 664 T + 1779618 T^{2} - 664 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.27632987659299655646781178799, −12.11050418120095162927090243564, −11.53596273464521670832837473914, −11.38877843987926079005848866597, −10.39527046502980608642461348633, −10.11273323659505284049996330443, −9.265470538002553362706371216565, −8.883230783940692852695745409819, −8.342258761951479821104636953239, −7.64659685835112797103145769696, −6.73961172359149170084649142515, −6.44846109697459173396428217355, −5.62749816620686311188033767704, −5.51343900559437254962157215947, −4.66444943384070729710683771265, −4.30097411355325161304113297813, −3.48890578831053268065585151651, −3.22541706504688024702033440526, −1.92515513340998566052728157358, −1.10424273914690404018298632997,
1.10424273914690404018298632997, 1.92515513340998566052728157358, 3.22541706504688024702033440526, 3.48890578831053268065585151651, 4.30097411355325161304113297813, 4.66444943384070729710683771265, 5.51343900559437254962157215947, 5.62749816620686311188033767704, 6.44846109697459173396428217355, 6.73961172359149170084649142515, 7.64659685835112797103145769696, 8.342258761951479821104636953239, 8.883230783940692852695745409819, 9.265470538002553362706371216565, 10.11273323659505284049996330443, 10.39527046502980608642461348633, 11.38877843987926079005848866597, 11.53596273464521670832837473914, 12.11050418120095162927090243564, 12.27632987659299655646781178799
