| L(s) = 1 | − 3·3-s − 14·5-s − 4·11-s + 108·13-s + 42·15-s + 14·17-s − 92·19-s + 152·23-s + 125·25-s + 27·27-s − 212·29-s + 144·31-s + 12·33-s − 158·37-s − 324·39-s − 780·41-s − 1.01e3·43-s + 528·47-s − 42·51-s − 606·53-s + 56·55-s + 276·57-s + 364·59-s − 678·61-s − 1.51e3·65-s − 844·67-s − 456·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.25·5-s − 0.109·11-s + 2.30·13-s + 0.722·15-s + 0.199·17-s − 1.11·19-s + 1.37·23-s + 25-s + 0.192·27-s − 1.35·29-s + 0.834·31-s + 0.0633·33-s − 0.702·37-s − 1.33·39-s − 2.97·41-s − 3.60·43-s + 1.63·47-s − 0.115·51-s − 1.57·53-s + 0.137·55-s + 0.641·57-s + 0.803·59-s − 1.42·61-s − 2.88·65-s − 1.53·67-s − 0.795·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.08417657229\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08417657229\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 14 T + 71 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T - 1315 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 54 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 14 T - 4717 T^{2} - 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 92 T + 1605 T^{2} + 92 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 152 T + 10937 T^{2} - 152 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 106 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 144 T - 9055 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 158 T - 25689 T^{2} + 158 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 390 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 508 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 528 T + 174961 T^{2} - 528 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 606 T + 218359 T^{2} + 606 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 364 T - 72883 T^{2} - 364 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 678 T + 232703 T^{2} + 678 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 844 T + 411573 T^{2} + 844 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 422 T - 210933 T^{2} - 422 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 384 T - 345583 T^{2} + 384 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 548 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 1194 T + 720667 T^{2} + 1194 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 1502 T + p^{3} T^{2} )^{2} \) |
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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
−10.71007958256415144953456624160, −10.22113591129320116489378126048, −9.747689147172184987445118173993, −8.946876767163782537914800335277, −8.629662634215789331548687531877, −8.327907883784595903889668106389, −8.109095036545003572351853855640, −7.24204085428413570417077202273, −6.87520399321193183965056400294, −6.52315648564995398553627961534, −6.04700679747593840765199914588, −5.32396337660189184897287211027, −5.05862910910488125454326118382, −4.28975681911657657082182059729, −3.89745433927964674163016213857, −3.27192605291385878575052164001, −3.02702136813274256252712982239, −1.56156187896232319924575176752, −1.38160132855859268099628637257, −0.093945551802059977620418330843,
0.093945551802059977620418330843, 1.38160132855859268099628637257, 1.56156187896232319924575176752, 3.02702136813274256252712982239, 3.27192605291385878575052164001, 3.89745433927964674163016213857, 4.28975681911657657082182059729, 5.05862910910488125454326118382, 5.32396337660189184897287211027, 6.04700679747593840765199914588, 6.52315648564995398553627961534, 6.87520399321193183965056400294, 7.24204085428413570417077202273, 8.109095036545003572351853855640, 8.327907883784595903889668106389, 8.629662634215789331548687531877, 8.946876767163782537914800335277, 9.747689147172184987445118173993, 10.22113591129320116489378126048, 10.71007958256415144953456624160
