| L(s) = 1 | + 2·2-s + 3·3-s + 8·5-s + 6·6-s − 8·8-s + 16·10-s − 40·11-s − 8·13-s + 24·15-s − 16·16-s − 84·17-s + 148·19-s − 80·22-s − 84·23-s − 24·24-s + 125·25-s − 16·26-s − 27·27-s + 116·29-s + 48·30-s − 136·31-s − 120·33-s − 168·34-s + 222·37-s + 296·38-s − 24·39-s − 64·40-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.715·5-s + 0.408·6-s − 0.353·8-s + 0.505·10-s − 1.09·11-s − 0.170·13-s + 0.413·15-s − 1/4·16-s − 1.19·17-s + 1.78·19-s − 0.775·22-s − 0.761·23-s − 0.204·24-s + 25-s − 0.120·26-s − 0.192·27-s + 0.742·29-s + 0.292·30-s − 0.787·31-s − 0.633·33-s − 0.847·34-s + 0.986·37-s + 1.26·38-s − 0.0985·39-s − 0.252·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.717666746\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.717666746\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 3 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T - 61 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 40 T + 269 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 84 T + 2143 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 148 T + 15045 T^{2} - 148 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 84 T - 5111 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 p T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 136 T - 11295 T^{2} + 136 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 6 p T - p^{2} T^{2} - 6 p^{4} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 420 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 164 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 488 T + 134321 T^{2} - 488 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 478 T + 79607 T^{2} + 478 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 548 T + 94925 T^{2} - 548 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 692 T + 251883 T^{2} - 692 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 908 T + 523701 T^{2} - 908 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 524 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 440 T - 195417 T^{2} - 440 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 1216 T + 985617 T^{2} + 1216 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 684 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 604 T - 340153 T^{2} - 604 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 832 T + p^{3} T^{2} )^{2} \) |
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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
−12.04337408867386439033780628599, −11.20457622797132890052127918915, −10.56840919428586153174500842100, −10.01619289046212902520901866324, −9.848887556166728628584299980364, −9.247897673003674172837649239320, −8.504385117273531726925665081633, −8.496396408539746296631705331261, −7.70955720489713746429190774881, −7.07996643640946536762094145651, −6.73213554757505860790080099249, −5.95225539949918315197987820063, −5.48309690172798292664055715438, −4.92702362444011381624477243819, −4.61870977771703405569369845131, −3.51672312441144610069428066530, −3.22220765963025643595596947335, −2.39722296237951101947406976905, −1.88634159504540447324488550967, −0.60213721238511986347467921084,
0.60213721238511986347467921084, 1.88634159504540447324488550967, 2.39722296237951101947406976905, 3.22220765963025643595596947335, 3.51672312441144610069428066530, 4.61870977771703405569369845131, 4.92702362444011381624477243819, 5.48309690172798292664055715438, 5.95225539949918315197987820063, 6.73213554757505860790080099249, 7.07996643640946536762094145651, 7.70955720489713746429190774881, 8.496396408539746296631705331261, 8.504385117273531726925665081633, 9.247897673003674172837649239320, 9.848887556166728628584299980364, 10.01619289046212902520901866324, 10.56840919428586153174500842100, 11.20457622797132890052127918915, 12.04337408867386439033780628599
