| L(s) = 1 | + 1.32·3-s − 1.97·5-s + 1.77·7-s − 1.23·9-s + 5.56·11-s − 13-s − 2.62·15-s − 1.80·17-s + 4.98·19-s + 2.35·21-s + 5.92·23-s − 1.10·25-s − 5.62·27-s + 29-s + 8.50·31-s + 7.39·33-s − 3.50·35-s + 2.17·37-s − 1.32·39-s + 2.97·41-s − 1.09·43-s + 2.43·45-s − 8.08·47-s − 3.85·49-s − 2.39·51-s − 11.1·53-s − 10.9·55-s + ⋯ |
| L(s) = 1 | + 0.767·3-s − 0.882·5-s + 0.670·7-s − 0.411·9-s + 1.67·11-s − 0.277·13-s − 0.677·15-s − 0.437·17-s + 1.14·19-s + 0.514·21-s + 1.23·23-s − 0.220·25-s − 1.08·27-s + 0.185·29-s + 1.52·31-s + 1.28·33-s − 0.591·35-s + 0.357·37-s − 0.212·39-s + 0.465·41-s − 0.166·43-s + 0.363·45-s − 1.17·47-s − 0.550·49-s − 0.335·51-s − 1.52·53-s − 1.48·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.609700960\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.609700960\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 1.32T + 3T^{2} \) |
| 5 | \( 1 + 1.97T + 5T^{2} \) |
| 7 | \( 1 - 1.77T + 7T^{2} \) |
| 11 | \( 1 - 5.56T + 11T^{2} \) |
| 17 | \( 1 + 1.80T + 17T^{2} \) |
| 19 | \( 1 - 4.98T + 19T^{2} \) |
| 23 | \( 1 - 5.92T + 23T^{2} \) |
| 31 | \( 1 - 8.50T + 31T^{2} \) |
| 37 | \( 1 - 2.17T + 37T^{2} \) |
| 41 | \( 1 - 2.97T + 41T^{2} \) |
| 43 | \( 1 + 1.09T + 43T^{2} \) |
| 47 | \( 1 + 8.08T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 - 2.03T + 59T^{2} \) |
| 61 | \( 1 + 5.00T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 + 1.28T + 71T^{2} \) |
| 73 | \( 1 - 3.36T + 73T^{2} \) |
| 79 | \( 1 - 5.84T + 79T^{2} \) |
| 83 | \( 1 - 0.0332T + 83T^{2} \) |
| 89 | \( 1 - 2.98T + 89T^{2} \) |
| 97 | \( 1 - 2.31T + 97T^{2} \) |
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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
−8.178231551140195379158891164209, −7.51981466959811232761691582126, −6.79622123199974703766893321184, −6.07044308906625531469117316757, −4.94035739784785929472125057810, −4.41215146207929480189061095738, −3.50309924235870538551456400958, −3.00840759355919028102349568043, −1.84763430556259651533645189714, −0.849660778802930960401262892447,
0.849660778802930960401262892447, 1.84763430556259651533645189714, 3.00840759355919028102349568043, 3.50309924235870538551456400958, 4.41215146207929480189061095738, 4.94035739784785929472125057810, 6.07044308906625531469117316757, 6.79622123199974703766893321184, 7.51981466959811232761691582126, 8.178231551140195379158891164209
