| L(s) = 1 | + 5·5-s + 28·7-s + 24·11-s + 70·13-s + 204·17-s + 40·19-s + 72·23-s − 306·29-s + 136·31-s + 140·35-s − 428·37-s + 150·41-s + 292·43-s + 72·47-s + 343·49-s − 828·53-s + 120·55-s + 744·59-s + 418·61-s + 350·65-s − 188·67-s + 960·71-s + 868·73-s + 672·77-s − 1.35e3·79-s + 612·83-s + 1.02e3·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s + 0.657·11-s + 1.49·13-s + 2.91·17-s + 0.482·19-s + 0.652·23-s − 1.95·29-s + 0.787·31-s + 0.676·35-s − 1.90·37-s + 0.571·41-s + 1.03·43-s + 0.223·47-s + 49-s − 2.14·53-s + 0.294·55-s + 1.64·59-s + 0.877·61-s + 0.667·65-s − 0.342·67-s + 1.60·71-s + 1.39·73-s + 0.994·77-s − 1.92·79-s + 0.809·83-s + 1.30·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.557315600\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.557315600\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 5 p T + p^{3} T^{2} )( 1 + p T + p^{3} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 24 T - 755 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 89 T + p^{3} T^{2} )( 1 + 19 T + p^{3} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 p T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 20 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 72 T - 6983 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 306 T + 69247 T^{2} + 306 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 136 T - 11295 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 214 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 150 T - 46421 T^{2} - 150 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 292 T + 5757 T^{2} - 292 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 72 T - 98639 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 414 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 744 T + 348157 T^{2} - 744 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 418 T - 52257 T^{2} - 418 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 188 T - 265419 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 480 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 434 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 1352 T + 1334865 T^{2} + 1352 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 612 T - 197243 T^{2} - 612 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 30 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 286 T - 830877 T^{2} - 286 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.058430558841057384794197740581, −9.004950288388443492818748560512, −8.273532226706615999655988689460, −8.038271361946161501970497748082, −7.80420787189111440276171276538, −7.28341902959714912689409168425, −6.91549750502524689844509254828, −6.32354982596190584996862678230, −5.80833882506336601245316338744, −5.59943108236760566824742908738, −5.11547745629316050356351775688, −4.95052222643614238358329786420, −3.93901412510655582070102294841, −3.84371858473859105265148134215, −3.33980812878794745722594568203, −2.81209350492732998515621274386, −1.84197804878582479252471801470, −1.64826094086859136519938954158, −1.05097088206094220398406702252, −0.75947991292065423044074001193,
0.75947991292065423044074001193, 1.05097088206094220398406702252, 1.64826094086859136519938954158, 1.84197804878582479252471801470, 2.81209350492732998515621274386, 3.33980812878794745722594568203, 3.84371858473859105265148134215, 3.93901412510655582070102294841, 4.95052222643614238358329786420, 5.11547745629316050356351775688, 5.59943108236760566824742908738, 5.80833882506336601245316338744, 6.32354982596190584996862678230, 6.91549750502524689844509254828, 7.28341902959714912689409168425, 7.80420787189111440276171276538, 8.038271361946161501970497748082, 8.273532226706615999655988689460, 9.004950288388443492818748560512, 9.058430558841057384794197740581
