| L(s) = 1 | − 16·4-s + 405·9-s − 374·11-s + 256·16-s − 516·19-s − 2.59e3·29-s + 3.83e3·31-s − 6.48e3·36-s + 1.33e4·41-s + 5.98e3·44-s − 2.40e3·49-s − 7.86e3·59-s − 9.74e4·61-s − 4.09e3·64-s + 1.27e5·71-s + 8.25e3·76-s + 8.74e4·79-s + 1.04e5·81-s − 9.11e4·89-s − 1.51e5·99-s − 8.99e4·101-s − 1.65e4·109-s + 4.15e4·116-s − 2.17e5·121-s − 6.13e4·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 5/3·9-s − 0.931·11-s + 1/4·16-s − 0.327·19-s − 0.573·29-s + 0.716·31-s − 5/6·36-s + 1.24·41-s + 0.465·44-s − 1/7·49-s − 0.294·59-s − 3.35·61-s − 1/8·64-s + 3.00·71-s + 0.163·76-s + 1.57·79-s + 16/9·81-s − 1.21·89-s − 1.55·99-s − 0.877·101-s − 0.133·109-s + 0.286·116-s − 1.34·121-s − 0.358·124-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.017096583\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.017096583\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 p^{4} T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 17 p T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 349457 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 447255 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 258 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 4051786 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 1299 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 1916 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 95417830 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6676 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 283917202 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 25353987 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 151626762 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 3932 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 48740 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 690341990 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 63736 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 489522286 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 43721 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 230010 p^{2} T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 45560 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 13891963489 T^{2} + p^{10} T^{4} \) |
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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
−11.05699028783161484141450211078, −10.47906732428098559445644187128, −9.800798676645035956057119740927, −9.665237736431242843906902789257, −9.145382413939012334237956031089, −8.586251534171562978272507600477, −7.80849775331908746972892981901, −7.77568503572756408848118064551, −7.24036024944951791779626823826, −6.39280228010322361253858911285, −6.30866954542946591471542613957, −5.20304712749982916182773508495, −5.11567945929523758931851025246, −4.17578634295150094482908847181, −4.16030512393228373237840730476, −3.23190278568680553878919085783, −2.53655569775089899785250632122, −1.79968063732603726623319593322, −1.14858579790761004906671429919, −0.40158329257998323542529266675,
0.40158329257998323542529266675, 1.14858579790761004906671429919, 1.79968063732603726623319593322, 2.53655569775089899785250632122, 3.23190278568680553878919085783, 4.16030512393228373237840730476, 4.17578634295150094482908847181, 5.11567945929523758931851025246, 5.20304712749982916182773508495, 6.30866954542946591471542613957, 6.39280228010322361253858911285, 7.24036024944951791779626823826, 7.77568503572756408848118064551, 7.80849775331908746972892981901, 8.586251534171562978272507600477, 9.145382413939012334237956031089, 9.665237736431242843906902789257, 9.800798676645035956057119740927, 10.47906732428098559445644187128, 11.05699028783161484141450211078
