| L(s) = 1 | − 14·7-s − 2·9-s − 4·11-s − 52·23-s − 5·25-s − 44·29-s + 28·37-s + 68·43-s + 147·49-s − 68·53-s + 28·63-s − 28·67-s − 124·71-s + 56·77-s − 76·79-s − 77·81-s + 8·99-s + 212·107-s − 284·109-s − 68·113-s − 230·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 2·7-s − 2/9·9-s − 0.363·11-s − 2.26·23-s − 1/5·25-s − 1.51·29-s + 0.756·37-s + 1.58·43-s + 3·49-s − 1.28·53-s + 4/9·63-s − 0.417·67-s − 1.74·71-s + 8/11·77-s − 0.962·79-s − 0.950·81-s + 8/99·99-s + 1.98·107-s − 2.60·109-s − 0.601·113-s − 1.90·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1160368554\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1160368554\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )( 1 + 4 T + p^{2} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 158 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 142 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 542 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 26 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 958 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2642 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 34 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 3698 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 34 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 5342 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 1378 T^{2} + p^{4} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 62 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 7778 T^{2} + p^{4} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 38 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 12158 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 15122 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 18098 T^{2} + p^{4} T^{4} \) |
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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
−10.63293182426061935263679722163, −10.14257304403491253082801767448, −10.04793462606254852248155938841, −9.410213748086958664638882805359, −9.154073451939256915553121998010, −8.770269462542081119546464760205, −7.88560447078840554153168128819, −7.74488605482507494585832078051, −7.23719304036575454984750404237, −6.56280472103517479617250412496, −6.17937658522725185502358640817, −5.81669745471889639274376696484, −5.48364209161872133506893961384, −4.52883190807371442812281487379, −3.80385629949481119769284773598, −3.78678625820937120452354907726, −2.73574460785639361706415047079, −2.55008658982862606074718514863, −1.48562548840913325532482908686, −0.12779721363889711470770518493,
0.12779721363889711470770518493, 1.48562548840913325532482908686, 2.55008658982862606074718514863, 2.73574460785639361706415047079, 3.78678625820937120452354907726, 3.80385629949481119769284773598, 4.52883190807371442812281487379, 5.48364209161872133506893961384, 5.81669745471889639274376696484, 6.17937658522725185502358640817, 6.56280472103517479617250412496, 7.23719304036575454984750404237, 7.74488605482507494585832078051, 7.88560447078840554153168128819, 8.770269462542081119546464760205, 9.154073451939256915553121998010, 9.410213748086958664638882805359, 10.04793462606254852248155938841, 10.14257304403491253082801767448, 10.63293182426061935263679722163
