| L(s) = 1 | − 7-s − 14·19-s + 10·25-s + 8·31-s − 22·37-s − 6·49-s + 26·103-s + 4·109-s + 22·121-s + 127-s + 131-s + 14·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s − 10·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 3.21·19-s + 2·25-s + 1.43·31-s − 3.61·37-s − 6/7·49-s + 2.56·103-s + 0.383·109-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 1.21·133-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 − 0.0769·169-s + 0.0760·173-s − 0.755·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8847238762\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8847238762\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−8.853081095465138699184656903214, −8.623476487465571274226712558019, −8.302491159645706900020484982475, −7.926677556188623053776491347935, −7.24510965040544038635997531694, −6.90577504287534565230776954988, −6.73050437750008015806638949079, −6.22026805230902556378040357776, −6.13898765654880443285069397969, −5.43459385225496633594064220032, −4.80798341981885920811238398123, −4.79514110234140824617061367210, −4.32589373637895868643257683056, −3.60095581417708514291194309067, −3.47203153013234921401297197867, −2.80670669404334344327325386113, −2.30259967256612242491342171503, −1.89662193330354235609065944471, −1.22956292702983013668428263382, −0.30031630577067337599482413903,
0.30031630577067337599482413903, 1.22956292702983013668428263382, 1.89662193330354235609065944471, 2.30259967256612242491342171503, 2.80670669404334344327325386113, 3.47203153013234921401297197867, 3.60095581417708514291194309067, 4.32589373637895868643257683056, 4.79514110234140824617061367210, 4.80798341981885920811238398123, 5.43459385225496633594064220032, 6.13898765654880443285069397969, 6.22026805230902556378040357776, 6.73050437750008015806638949079, 6.90577504287534565230776954988, 7.24510965040544038635997531694, 7.926677556188623053776491347935, 8.302491159645706900020484982475, 8.623476487465571274226712558019, 8.853081095465138699184656903214
