| L(s) = 1 | − 8.79e12·2-s + 8.97e20·3-s + 7.73e25·4-s + 3.21e30·5-s − 7.89e33·6-s + 8.51e36·7-s − 6.80e38·8-s + 4.82e41·9-s − 2.82e43·10-s + 1.44e45·11-s + 6.94e46·12-s + 1.58e48·13-s − 7.48e49·14-s + 2.88e51·15-s + 5.98e51·16-s − 6.18e53·17-s − 4.24e54·18-s + 6.29e55·19-s + 2.48e56·20-s + 7.64e57·21-s − 1.26e58·22-s + 2.79e58·23-s − 6.11e59·24-s + 3.88e60·25-s − 1.38e61·26-s + 1.43e62·27-s + 6.58e62·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.57·3-s + 0.5·4-s + 1.26·5-s − 1.11·6-s + 1.47·7-s − 0.353·8-s + 1.49·9-s − 0.894·10-s + 0.721·11-s + 0.789·12-s + 0.552·13-s − 1.04·14-s + 1.99·15-s + 0.250·16-s − 1.84·17-s − 1.05·18-s + 1.48·19-s + 0.632·20-s + 2.32·21-s − 0.510·22-s + 0.162·23-s − 0.558·24-s + 0.601·25-s − 0.390·26-s + 0.779·27-s + 0.736·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(88-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+87/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(44)\) |
\(\approx\) |
\(5.012219514\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.012219514\) |
| \(L(\frac{89}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8.79e12T \) |
| good | 3 | \( 1 - 8.97e20T + 3.23e41T^{2} \) |
| 5 | \( 1 - 3.21e30T + 6.46e60T^{2} \) |
| 7 | \( 1 - 8.51e36T + 3.33e73T^{2} \) |
| 11 | \( 1 - 1.44e45T + 3.99e90T^{2} \) |
| 13 | \( 1 - 1.58e48T + 8.18e96T^{2} \) |
| 17 | \( 1 + 6.18e53T + 1.11e107T^{2} \) |
| 19 | \( 1 - 6.29e55T + 1.78e111T^{2} \) |
| 23 | \( 1 - 2.79e58T + 2.95e118T^{2} \) |
| 29 | \( 1 - 3.72e63T + 1.69e127T^{2} \) |
| 31 | \( 1 - 9.26e64T + 5.60e129T^{2} \) |
| 37 | \( 1 - 5.17e67T + 2.71e136T^{2} \) |
| 41 | \( 1 + 6.26e69T + 2.05e140T^{2} \) |
| 43 | \( 1 + 1.26e71T + 1.29e142T^{2} \) |
| 47 | \( 1 + 8.18e72T + 2.96e145T^{2} \) |
| 53 | \( 1 + 1.68e75T + 1.02e150T^{2} \) |
| 59 | \( 1 + 1.48e77T + 1.15e154T^{2} \) |
| 61 | \( 1 + 3.28e77T + 2.10e155T^{2} \) |
| 67 | \( 1 - 3.12e79T + 7.38e158T^{2} \) |
| 71 | \( 1 + 2.96e80T + 1.14e161T^{2} \) |
| 73 | \( 1 - 9.74e80T + 1.28e162T^{2} \) |
| 79 | \( 1 + 4.07e82T + 1.24e165T^{2} \) |
| 83 | \( 1 - 4.93e82T + 9.11e166T^{2} \) |
| 89 | \( 1 + 4.22e83T + 3.95e169T^{2} \) |
| 97 | \( 1 + 3.85e86T + 7.06e172T^{2} \) |
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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
−13.56212738667924720320814317694, −11.34531231468597100989383656280, −9.788579068578735213121355107679, −8.869507221309614663126915566889, −8.041493609046368949896711463115, −6.54249269705281997050589741877, −4.68886295846973120643187208938, −2.97141330193538226762185766685, −1.84829865759006224261982024663, −1.35574022902466705355234506159,
1.35574022902466705355234506159, 1.84829865759006224261982024663, 2.97141330193538226762185766685, 4.68886295846973120643187208938, 6.54249269705281997050589741877, 8.041493609046368949896711463115, 8.869507221309614663126915566889, 9.788579068578735213121355107679, 11.34531231468597100989383656280, 13.56212738667924720320814317694
