| L(s) = 1 | + 4·4-s + 4·5-s + 5·9-s − 2·11-s + 12·16-s − 4·19-s + 16·20-s + 11·25-s − 14·29-s + 8·31-s + 20·36-s − 12·41-s − 8·44-s + 20·45-s − 49-s − 8·55-s + 8·59-s − 20·61-s + 32·64-s − 16·76-s + 30·79-s + 48·80-s + 16·81-s − 12·89-s − 16·95-s − 10·99-s + 44·100-s + ⋯ |
| L(s) = 1 | + 2·4-s + 1.78·5-s + 5/3·9-s − 0.603·11-s + 3·16-s − 0.917·19-s + 3.57·20-s + 11/5·25-s − 2.59·29-s + 1.43·31-s + 10/3·36-s − 1.87·41-s − 1.20·44-s + 2.98·45-s − 1/7·49-s − 1.07·55-s + 1.04·59-s − 2.56·61-s + 4·64-s − 1.83·76-s + 3.37·79-s + 5.36·80-s + 16/9·81-s − 1.27·89-s − 1.64·95-s − 1.00·99-s + 22/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.535390306\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.535390306\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| 23 | $C_2$ | \( 1 + T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
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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
−10.33939717377496911447816131480, −10.28264913001339705097655773335, −9.639809411114690986172689372894, −9.558859801260379362016699305834, −8.847315227707728203639687128229, −8.225554146746851270178060743813, −7.66610918887779224527449920412, −7.46298483827374635670017174123, −6.79123531601681213051097260453, −6.64166048917523487530186401781, −6.18269645525932503457314371327, −5.85746103593950652600421213696, −5.10503983165480828949327573623, −4.98438399121637655439064877371, −3.92096839390185421544209138749, −3.45811102431292640504735563808, −2.56003569336734545883504796702, −2.31876623380178061718467672047, −1.59907497627494386625249186578, −1.43608965076060847863653409644,
1.43608965076060847863653409644, 1.59907497627494386625249186578, 2.31876623380178061718467672047, 2.56003569336734545883504796702, 3.45811102431292640504735563808, 3.92096839390185421544209138749, 4.98438399121637655439064877371, 5.10503983165480828949327573623, 5.85746103593950652600421213696, 6.18269645525932503457314371327, 6.64166048917523487530186401781, 6.79123531601681213051097260453, 7.46298483827374635670017174123, 7.66610918887779224527449920412, 8.225554146746851270178060743813, 8.847315227707728203639687128229, 9.558859801260379362016699305834, 9.639809411114690986172689372894, 10.28264913001339705097655773335, 10.33939717377496911447816131480
