| L(s) = 1 | − 2-s + 7-s + 8-s + 9-s − 11-s − 14-s − 16-s − 18-s + 22-s − 23-s − 29-s − 37-s − 43-s + 46-s + 49-s + 2·53-s + 56-s + 58-s + 63-s + 64-s − 67-s − 71-s + 72-s + 74-s − 77-s − 79-s + 81-s + ⋯ |
| L(s) = 1 | − 2-s + 7-s + 8-s + 9-s − 11-s − 14-s − 16-s − 18-s + 22-s − 23-s − 29-s − 37-s − 43-s + 46-s + 49-s + 2·53-s + 56-s + 58-s + 63-s + 64-s − 67-s − 71-s + 72-s + 74-s − 77-s − 79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4347380175\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4347380175\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 + T + T^{2} \) |
| 3 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 + T + T^{2} \) |
| 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
−13.03670795456117638900866772070, −11.77331474767626381219807458502, −10.58543688988728015975855782152, −10.06503499894156669614475940200, −8.846864120040407752443865863336, −7.928371315358622709755528775163, −7.21219424452815957631789646245, −5.32772362441185835516557683036, −4.20961109287914824166132706629, −1.81080441085222069438971480503,
1.81080441085222069438971480503, 4.20961109287914824166132706629, 5.32772362441185835516557683036, 7.21219424452815957631789646245, 7.928371315358622709755528775163, 8.846864120040407752443865863336, 10.06503499894156669614475940200, 10.58543688988728015975855782152, 11.77331474767626381219807458502, 13.03670795456117638900866772070
