| L(s) = 1 | + 7·4-s − 10·9-s − 48·11-s − 15·16-s − 232·19-s − 60·29-s − 344·31-s − 70·36-s − 684·41-s − 336·44-s − 98·49-s − 504·59-s + 220·61-s − 553·64-s − 1.41e3·71-s − 1.62e3·76-s + 968·79-s − 629·81-s + 1.54e3·89-s + 480·99-s − 420·101-s + 2.37e3·109-s − 420·116-s − 934·121-s − 2.40e3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 7/8·4-s − 0.370·9-s − 1.31·11-s − 0.234·16-s − 2.80·19-s − 0.384·29-s − 1.99·31-s − 0.324·36-s − 2.60·41-s − 1.15·44-s − 2/7·49-s − 1.11·59-s + 0.461·61-s − 1.08·64-s − 2.36·71-s − 2.45·76-s + 1.37·79-s − 0.862·81-s + 1.84·89-s + 0.487·99-s − 0.413·101-s + 2.08·109-s − 0.336·116-s − 0.701·121-s − 1.74·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 17 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 7 T^{2} + p^{6} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 10 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 p^{2} T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 24 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 1030 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 116 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 20734 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 30 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 172 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 97942 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 342 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 137110 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 124702 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 p^{2} T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 252 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 110 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 367270 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 708 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 646990 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 484 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 572038 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 774 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1679422 T^{2} + p^{6} T^{4} \) |
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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
−10.48262812718079295191751081100, −10.45003594599013903395813572461, −9.748953406935180400184091027290, −8.916466472894863703065734888805, −8.772156140594064990213604724660, −8.317040883954338335083503122259, −7.52430748825825200745536874118, −7.48714667875867588271705789486, −6.61879848170905525825960160666, −6.41697704808275461280133467235, −5.84798190277569519804114226122, −5.21524494767023274350368859995, −4.76542088483137001492988901804, −4.06550794064444127682719866466, −3.37760450585846030072698215390, −2.73931685831830878603720113640, −2.00323208346037421144177330888, −1.81424147318915866618309580227, 0, 0,
1.81424147318915866618309580227, 2.00323208346037421144177330888, 2.73931685831830878603720113640, 3.37760450585846030072698215390, 4.06550794064444127682719866466, 4.76542088483137001492988901804, 5.21524494767023274350368859995, 5.84798190277569519804114226122, 6.41697704808275461280133467235, 6.61879848170905525825960160666, 7.48714667875867588271705789486, 7.52430748825825200745536874118, 8.317040883954338335083503122259, 8.772156140594064990213604724660, 8.916466472894863703065734888805, 9.748953406935180400184091027290, 10.45003594599013903395813572461, 10.48262812718079295191751081100
