| L(s) = 1 | + 2·5-s − 7-s + 3·11-s + 2·13-s + 9·17-s + 8·19-s − 9·23-s + 3·25-s + 6·29-s + 2·31-s − 2·35-s + 37-s + 3·41-s − 10·43-s − 5·49-s + 9·53-s + 6·55-s − 6·59-s + 19·61-s + 4·65-s + 8·67-s + 3·71-s − 8·73-s − 3·77-s + 5·79-s − 18·83-s + 18·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.377·7-s + 0.904·11-s + 0.554·13-s + 2.18·17-s + 1.83·19-s − 1.87·23-s + 3/5·25-s + 1.11·29-s + 0.359·31-s − 0.338·35-s + 0.164·37-s + 0.468·41-s − 1.52·43-s − 5/7·49-s + 1.23·53-s + 0.809·55-s − 0.781·59-s + 2.43·61-s + 0.496·65-s + 0.977·67-s + 0.356·71-s − 0.936·73-s − 0.341·77-s + 0.562·79-s − 1.97·83-s + 1.95·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.849944250\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.849944250\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 3 T + 16 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 9 T + 46 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 76 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 78 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 9 T + 118 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 94 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 19 T + 204 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 3 T + 136 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 5 T - 42 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 18 T + 214 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 9 T + 124 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 29 T + 396 T^{2} + 29 p T^{3} + p^{2} T^{4} \) |
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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
−7.77101989528550341770047707732, −7.72927398809971664588979105047, −7.00591897768724380907370697029, −6.91141796006616364677409476381, −6.35851277295490706870064120922, −6.30238160308665860766059054056, −5.70908495849140922418411518502, −5.55954538757103709283753768432, −5.28743321405102824054128997708, −4.91653947876554260280886616552, −4.27228604596570394757109713647, −3.98890704159087690979832426438, −3.48185027999375245305525985248, −3.39344704474328219154890807134, −2.76975286392117622157820202433, −2.57108207479084605908880228301, −1.71926944566319832974660438042, −1.54898459005358564087585722476, −0.994979197858359544550419001509, −0.63433802280451236892629522708,
0.63433802280451236892629522708, 0.994979197858359544550419001509, 1.54898459005358564087585722476, 1.71926944566319832974660438042, 2.57108207479084605908880228301, 2.76975286392117622157820202433, 3.39344704474328219154890807134, 3.48185027999375245305525985248, 3.98890704159087690979832426438, 4.27228604596570394757109713647, 4.91653947876554260280886616552, 5.28743321405102824054128997708, 5.55954538757103709283753768432, 5.70908495849140922418411518502, 6.30238160308665860766059054056, 6.35851277295490706870064120922, 6.91141796006616364677409476381, 7.00591897768724380907370697029, 7.72927398809971664588979105047, 7.77101989528550341770047707732
