| L(s) = 1 | − 2·3-s + 2·5-s − 2·7-s + 3·9-s − 4·11-s − 4·15-s + 4·17-s − 4·19-s + 4·21-s + 3·25-s − 4·27-s + 4·29-s − 4·31-s + 8·33-s − 4·35-s − 4·37-s + 4·41-s − 8·43-s + 6·45-s + 8·47-s + 3·49-s − 8·51-s + 8·53-s − 8·55-s + 8·57-s − 4·61-s − 6·63-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s − 0.755·7-s + 9-s − 1.20·11-s − 1.03·15-s + 0.970·17-s − 0.917·19-s + 0.872·21-s + 3/5·25-s − 0.769·27-s + 0.742·29-s − 0.718·31-s + 1.39·33-s − 0.676·35-s − 0.657·37-s + 0.624·41-s − 1.21·43-s + 0.894·45-s + 1.16·47-s + 3/7·49-s − 1.12·51-s + 1.09·53-s − 1.07·55-s + 1.05·57-s − 0.512·61-s − 0.755·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45158400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45158400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.495135653\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.495135653\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 11 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 110 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 70 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 150 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 16 T + 182 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 32 T + 438 T^{2} - 32 p T^{3} + p^{2} T^{4} \) |
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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
−8.006461465904903730308920570108, −7.87912209123383949180672900502, −7.17674344046504570419333261124, −7.15827243574664600385886153504, −6.62256569861893992782364972597, −6.40319569940687238379021658623, −6.01286381659231342896213378071, −5.58422495569700924072142244495, −5.44143638211443872388101867897, −5.20567834753200327901305140518, −4.50754197579746334485473486156, −4.49630783671953872135213074443, −3.64139814778658411064197989152, −3.53731082727911860070110246583, −2.85766804394961758494189301491, −2.50779590907838222623652671178, −2.02237042538272829921922962690, −1.58071059050447086514520473835, −0.816751008332718117402865207480, −0.42317207280104362063502615286,
0.42317207280104362063502615286, 0.816751008332718117402865207480, 1.58071059050447086514520473835, 2.02237042538272829921922962690, 2.50779590907838222623652671178, 2.85766804394961758494189301491, 3.53731082727911860070110246583, 3.64139814778658411064197989152, 4.49630783671953872135213074443, 4.50754197579746334485473486156, 5.20567834753200327901305140518, 5.44143638211443872388101867897, 5.58422495569700924072142244495, 6.01286381659231342896213378071, 6.40319569940687238379021658623, 6.62256569861893992782364972597, 7.15827243574664600385886153504, 7.17674344046504570419333261124, 7.87912209123383949180672900502, 8.006461465904903730308920570108
