| L(s) = 1 | − 8·7-s + 12·17-s + 6·25-s − 8·31-s + 4·41-s + 16·47-s + 34·49-s + 32·71-s + 12·73-s − 8·79-s + 20·89-s − 28·97-s − 24·103-s − 4·113-s − 96·119-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + ⋯ |
| L(s) = 1 | − 3.02·7-s + 2.91·17-s + 6/5·25-s − 1.43·31-s + 0.624·41-s + 2.33·47-s + 34/7·49-s + 3.79·71-s + 1.40·73-s − 0.900·79-s + 2.11·89-s − 2.84·97-s − 2.36·103-s − 0.376·113-s − 8.80·119-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.829633351\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.829633351\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−9.261463209821477941125778635800, −9.196157733225529089312465358172, −8.236843162783280849131552804236, −8.206271792843165666601521568412, −7.54098490973968202557563080677, −7.16715810554002748224133427983, −6.77209989029074272953434809146, −6.69180965880791105959217480226, −5.95904907682833684919655114106, −5.67422571615892971359132728138, −5.53859079119232906435950130673, −4.97782031131378739858302163881, −4.02882779336633236721320512629, −3.87076273697347354838757659673, −3.26844259369493471845001666673, −3.17169361119004245545314740202, −2.70317677274366157697142455996, −1.99818597084615958384961295381, −0.888605342906179785623422702376, −0.62769668396498882839898116695,
0.62769668396498882839898116695, 0.888605342906179785623422702376, 1.99818597084615958384961295381, 2.70317677274366157697142455996, 3.17169361119004245545314740202, 3.26844259369493471845001666673, 3.87076273697347354838757659673, 4.02882779336633236721320512629, 4.97782031131378739858302163881, 5.53859079119232906435950130673, 5.67422571615892971359132728138, 5.95904907682833684919655114106, 6.69180965880791105959217480226, 6.77209989029074272953434809146, 7.16715810554002748224133427983, 7.54098490973968202557563080677, 8.206271792843165666601521568412, 8.236843162783280849131552804236, 9.196157733225529089312465358172, 9.261463209821477941125778635800
