| L(s) = 1 | − 16·4-s − 24·11-s + 256·16-s + 2.12e3·19-s − 1.57e4·29-s + 1.03e4·31-s − 7.28e3·41-s + 384·44-s + 3.25e4·49-s − 2.86e4·59-s − 9.58e4·61-s − 4.09e3·64-s − 1.03e5·71-s − 3.39e4·76-s − 5.03e4·79-s − 1.51e5·89-s − 1.77e5·101-s − 3.62e4·109-s + 2.52e5·116-s − 3.21e5·121-s − 1.66e5·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.0598·11-s + 1/4·16-s + 1.34·19-s − 3.48·29-s + 1.94·31-s − 0.676·41-s + 0.0299·44-s + 1.93·49-s − 1.07·59-s − 3.29·61-s − 1/8·64-s − 2.44·71-s − 0.673·76-s − 0.907·79-s − 2.02·89-s − 1.72·101-s − 0.291·109-s + 1.74·116-s − 1.99·121-s − 0.970·124-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.07053084156\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07053084156\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 32590 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 12 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 718870 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 6910 p^{2} T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 1060 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 4969490 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 7890 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5192 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 129682010 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 3642 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 65523430 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 97892450 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 578017510 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 14340 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 47938 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 1605169750 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 51912 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 4001518510 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 25160 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 6596727670 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 75510 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 15224751550 T^{2} + p^{10} T^{4} \) |
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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
−10.78726855520332014540801123343, −9.969032478351885448809075079782, −9.576628099988045446863954518411, −9.137016443583774087552898915968, −8.900008099286484774872361142019, −8.223451748655338395694073579651, −7.64413347013860067126115027179, −7.44845839939376834691830498474, −6.97810192936610109666276224309, −6.06377910857388349559407130954, −5.87116454170814418918576109204, −5.30418448313549801016996393356, −4.78517523853562835039926027581, −4.15837645959130683235872925855, −3.73765720008763987763661476008, −3.00935145154580024072278212848, −2.59010208727547209907133030488, −1.46021803179062292323711403498, −1.31691786799994157736722790718, −0.06518207256015517363266000786,
0.06518207256015517363266000786, 1.31691786799994157736722790718, 1.46021803179062292323711403498, 2.59010208727547209907133030488, 3.00935145154580024072278212848, 3.73765720008763987763661476008, 4.15837645959130683235872925855, 4.78517523853562835039926027581, 5.30418448313549801016996393356, 5.87116454170814418918576109204, 6.06377910857388349559407130954, 6.97810192936610109666276224309, 7.44845839939376834691830498474, 7.64413347013860067126115027179, 8.223451748655338395694073579651, 8.900008099286484774872361142019, 9.137016443583774087552898915968, 9.576628099988045446863954518411, 9.969032478351885448809075079782, 10.78726855520332014540801123343
