| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 13-s − 14-s + 16-s + 17-s + 23-s + 25-s + 26-s + 28-s + 29-s − 31-s − 32-s − 34-s − 2·41-s − 43-s − 46-s + 49-s − 50-s − 52-s + 53-s − 56-s − 58-s + 59-s + 2·61-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 13-s − 14-s + 16-s + 17-s + 23-s + 25-s + 26-s + 28-s + 29-s − 31-s − 32-s − 34-s − 2·41-s − 43-s − 46-s + 49-s − 50-s − 52-s + 53-s − 56-s − 58-s + 59-s + 2·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8095573329\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8095573329\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 + T )^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( 1 - T + T^{2} \) |
| 61 | \( ( 1 - T )^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 + T )^{2} \) |
| 89 | \( 1 - T + T^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.750643966672921471953154273707, −8.664191360235917744395649637337, −8.301526262384710623106853650446, −7.25362798030586643970491810829, −6.87926158098828170409240684387, −5.49646380923950997989719765687, −4.90261392942102129869788172941, −3.40129930843034292577983456743, −2.34654157073656844742338874000, −1.19142370224948957587992909629,
1.19142370224948957587992909629, 2.34654157073656844742338874000, 3.40129930843034292577983456743, 4.90261392942102129869788172941, 5.49646380923950997989719765687, 6.87926158098828170409240684387, 7.25362798030586643970491810829, 8.301526262384710623106853650446, 8.664191360235917744395649637337, 9.750643966672921471953154273707
