| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 12-s + 13-s − 14-s + 16-s + 17-s + 19-s + 21-s + 23-s − 24-s − 26-s − 27-s + 28-s − 29-s − 32-s − 34-s − 2·37-s − 38-s + 39-s − 42-s − 46-s − 2·47-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 12-s + 13-s − 14-s + 16-s + 17-s + 19-s + 21-s + 23-s − 24-s − 26-s − 27-s + 28-s − 29-s − 32-s − 34-s − 2·37-s − 38-s + 39-s − 42-s − 46-s − 2·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.401917503\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.401917503\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - T + T^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 + T )^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 + T )^{2} \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 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
−8.580473968794107420689996442455, −8.118946616793202878208257373851, −7.54401872738220241771356261403, −6.81070882361566489926307553411, −5.71297665805286461169795068204, −5.09705589240384806246209216990, −3.55752583951670443004739169255, −3.19485769321295354061950124737, −1.98388522002088213427789411834, −1.26627674389081139921670943886,
1.26627674389081139921670943886, 1.98388522002088213427789411834, 3.19485769321295354061950124737, 3.55752583951670443004739169255, 5.09705589240384806246209216990, 5.71297665805286461169795068204, 6.81070882361566489926307553411, 7.54401872738220241771356261403, 8.118946616793202878208257373851, 8.580473968794107420689996442455
