| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 12-s + 16-s + 18-s − 24-s − 27-s − 2·31-s + 32-s + 36-s − 48-s − 49-s − 2·53-s − 54-s − 2·62-s + 64-s + 72-s − 2·79-s + 81-s + 2·83-s + 2·93-s − 96-s − 98-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 12-s + 16-s + 18-s − 24-s − 27-s − 2·31-s + 32-s + 36-s − 48-s − 49-s − 2·53-s − 54-s − 2·62-s + 64-s + 72-s − 2·79-s + 81-s + 2·83-s + 2·93-s − 96-s − 98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.273554570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.273554570\) |
| \(L(1)\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( ( 1 + T )^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 + T )^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( ( 1 + T )^{2} \) |
| 83 | \( ( 1 - T )^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( 1 + T^{2} \) |
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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
−11.10094230083755156059502219900, −10.38815368257517062373664104303, −9.364452786023179427443886781276, −7.85601418807906888489630287406, −7.01717816097987807121640673780, −6.16414784739005877681783732895, −5.35084943067174626418158945751, −4.49767228655233910498716040829, −3.40892457339190791930215249053, −1.76701698785465158745217666201,
1.76701698785465158745217666201, 3.40892457339190791930215249053, 4.49767228655233910498716040829, 5.35084943067174626418158945751, 6.16414784739005877681783732895, 7.01717816097987807121640673780, 7.85601418807906888489630287406, 9.364452786023179427443886781276, 10.38815368257517062373664104303, 11.10094230083755156059502219900
