| L(s) = 1 | + 140·5-s − 2.75e4·13-s − 3.39e4·17-s − 1.41e5·25-s + 6.83e4·29-s − 7.04e4·37-s + 9.69e5·41-s − 4.02e5·49-s + 1.70e6·53-s − 1.43e5·61-s − 3.85e6·65-s + 7.82e6·73-s − 4.75e6·85-s + 5.02e6·89-s − 1.00e5·97-s + 3.00e7·101-s − 3.56e7·109-s + 3.76e7·113-s + 1.35e7·121-s − 3.14e7·125-s + 127-s + 131-s + 137-s + 139-s + 9.57e6·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 0.500·5-s − 3.47·13-s − 1.67·17-s − 1.81·25-s + 0.520·29-s − 0.228·37-s + 2.19·41-s − 0.489·49-s + 1.57·53-s − 0.0808·61-s − 1.73·65-s + 2.35·73-s − 0.840·85-s + 0.755·89-s − 0.0111·97-s + 2.90·101-s − 2.63·109-s + 2.45·113-s + 0.697·121-s − 1.43·125-s + 0.260·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.249108428\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.249108428\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 14 p T + p^{7} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 402926 T^{2} + p^{14} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 13591418 T^{2} + p^{14} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 13758 T + p^{7} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 16994 T + p^{7} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 630363638 T^{2} + p^{14} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 5763312334 T^{2} + p^{14} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 34190 T + p^{7} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 40513345982 T^{2} + p^{14} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 35206 T + p^{7} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 484550 T + p^{7} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 91999366214 T^{2} + p^{14} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 443321730914 T^{2} + p^{14} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 851702 T + p^{7} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 4493632970278 T^{2} + p^{14} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 71630 T + p^{7} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 12026991155606 T^{2} + p^{14} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 17616631546222 T^{2} + p^{14} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 3912042 T + p^{7} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 38308877392478 T^{2} + p^{14} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 51911300603254 T^{2} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2510630 T + p^{7} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 50094 T + p^{7} T^{2} )^{2} \) |
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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
−9.688171789499171983292791275676, −9.635553287807251939272973728372, −9.012734463520527891286001363014, −8.633394137288917810448354746971, −7.75366717806810911995849160849, −7.69712399093094499699774142583, −7.19097055047325828793769956509, −6.74857262426180066678383783482, −6.22399685420671556808948699284, −5.74596091131043241525833969537, −4.99113150725758951516572582871, −4.97286811630157737194225012946, −4.26990212568829339819431069503, −3.92546288827085122967955446102, −2.93982695761566154176872324761, −2.44704813934288189802961221510, −2.16233832033697193121616100693, −1.84327089762561242554605097446, −0.50125889429309698600952441802, −0.49982941655933024192789923343,
0.49982941655933024192789923343, 0.50125889429309698600952441802, 1.84327089762561242554605097446, 2.16233832033697193121616100693, 2.44704813934288189802961221510, 2.93982695761566154176872324761, 3.92546288827085122967955446102, 4.26990212568829339819431069503, 4.97286811630157737194225012946, 4.99113150725758951516572582871, 5.74596091131043241525833969537, 6.22399685420671556808948699284, 6.74857262426180066678383783482, 7.19097055047325828793769956509, 7.69712399093094499699774142583, 7.75366717806810911995849160849, 8.633394137288917810448354746971, 9.012734463520527891286001363014, 9.635553287807251939272973728372, 9.688171789499171983292791275676
