| L(s) = 1 | + 2·3-s − 2·5-s + 3·9-s − 4·13-s − 4·15-s − 25-s + 4·27-s − 16·31-s + 4·37-s − 8·39-s + 12·41-s − 8·43-s − 6·45-s + 10·49-s − 12·53-s + 8·65-s + 8·71-s − 2·75-s + 16·79-s + 5·81-s + 24·83-s + 28·89-s − 32·93-s + 24·107-s + 8·111-s − 12·117-s − 14·121-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 9-s − 1.10·13-s − 1.03·15-s − 1/5·25-s + 0.769·27-s − 2.87·31-s + 0.657·37-s − 1.28·39-s + 1.87·41-s − 1.21·43-s − 0.894·45-s + 10/7·49-s − 1.64·53-s + 0.992·65-s + 0.949·71-s − 0.230·75-s + 1.80·79-s + 5/9·81-s + 2.63·83-s + 2.96·89-s − 3.31·93-s + 2.32·107-s + 0.759·111-s − 1.10·117-s − 1.27·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.066658788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.066658788\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.829584486892682325279613103536, −8.134515067791440563334750344702, −7.84370602085598289035840888683, −7.65942778005623696973776287030, −7.39868286171354696447874837833, −7.07956395228963760949152284517, −6.56079091901228065161528580486, −6.09324126502811829392224599480, −5.77664470785569002585189088426, −4.99559030762216562900428125268, −4.91651765509695897031941335120, −4.52377631543394273727610367039, −3.69398762171163444952696398128, −3.68002430109632250045186902585, −3.48562117117840399441614015921, −2.65760611288065635864675629291, −2.13026047433678039284921860868, −2.12324509069957569588633335662, −1.16754944604682028026514655090, −0.40164717097590757484998935695,
0.40164717097590757484998935695, 1.16754944604682028026514655090, 2.12324509069957569588633335662, 2.13026047433678039284921860868, 2.65760611288065635864675629291, 3.48562117117840399441614015921, 3.68002430109632250045186902585, 3.69398762171163444952696398128, 4.52377631543394273727610367039, 4.91651765509695897031941335120, 4.99559030762216562900428125268, 5.77664470785569002585189088426, 6.09324126502811829392224599480, 6.56079091901228065161528580486, 7.07956395228963760949152284517, 7.39868286171354696447874837833, 7.65942778005623696973776287030, 7.84370602085598289035840888683, 8.134515067791440563334750344702, 8.829584486892682325279613103536
