L(s) = 1 | − 0.193·2-s − 2.50·3-s − 1.96·4-s + 0.483·6-s + 0.166·7-s + 0.766·8-s + 3.25·9-s − 0.120·11-s + 4.90·12-s − 4.23·13-s − 0.0320·14-s + 3.77·16-s − 5.01·17-s − 0.629·18-s − 2.77·19-s − 0.415·21-s + 0.0233·22-s − 5.34·23-s − 1.91·24-s + 0.819·26-s − 0.645·27-s − 0.325·28-s + 0.290·29-s + 3.57·31-s − 2.26·32-s + 0.301·33-s + 0.970·34-s + ⋯ |
L(s) = 1 | − 0.136·2-s − 1.44·3-s − 0.981·4-s + 0.197·6-s + 0.0627·7-s + 0.270·8-s + 1.08·9-s − 0.0363·11-s + 1.41·12-s − 1.17·13-s − 0.00857·14-s + 0.944·16-s − 1.21·17-s − 0.148·18-s − 0.637·19-s − 0.0906·21-s + 0.00496·22-s − 1.11·23-s − 0.391·24-s + 0.160·26-s − 0.124·27-s − 0.0615·28-s + 0.0538·29-s + 0.641·31-s − 0.399·32-s + 0.0524·33-s + 0.166·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1305424587\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1305424587\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 + 0.193T + 2T^{2} \) |
| 3 | \( 1 + 2.50T + 3T^{2} \) |
| 7 | \( 1 - 0.166T + 7T^{2} \) |
| 11 | \( 1 + 0.120T + 11T^{2} \) |
| 13 | \( 1 + 4.23T + 13T^{2} \) |
| 17 | \( 1 + 5.01T + 17T^{2} \) |
| 19 | \( 1 + 2.77T + 19T^{2} \) |
| 23 | \( 1 + 5.34T + 23T^{2} \) |
| 29 | \( 1 - 0.290T + 29T^{2} \) |
| 31 | \( 1 - 3.57T + 31T^{2} \) |
| 37 | \( 1 + 0.104T + 37T^{2} \) |
| 41 | \( 1 - 7.23T + 41T^{2} \) |
| 43 | \( 1 + 6.76T + 43T^{2} \) |
| 47 | \( 1 + 1.90T + 47T^{2} \) |
| 53 | \( 1 - 1.56T + 53T^{2} \) |
| 59 | \( 1 + 6.34T + 59T^{2} \) |
| 61 | \( 1 + 0.0566T + 61T^{2} \) |
| 67 | \( 1 + 6.47T + 67T^{2} \) |
| 71 | \( 1 + 9.47T + 71T^{2} \) |
| 73 | \( 1 + 15.6T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 + 4.84T + 83T^{2} \) |
| 89 | \( 1 + 5.92T + 89T^{2} \) |
| 97 | \( 1 - 4.77T + 97T^{2} \) |
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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
−8.082481163509190902275421092553, −7.31265547647737000073013401740, −6.47235050128768789166164858955, −5.94477508735196408552280784869, −5.10224800072260849315662237913, −4.59067728854369615504110622106, −4.09354139226530300077860541284, −2.72674747529396413553153869314, −1.54667391257635251053096928635, −0.21330483999755900099387615792,
0.21330483999755900099387615792, 1.54667391257635251053096928635, 2.72674747529396413553153869314, 4.09354139226530300077860541284, 4.59067728854369615504110622106, 5.10224800072260849315662237913, 5.94477508735196408552280784869, 6.47235050128768789166164858955, 7.31265547647737000073013401740, 8.082481163509190902275421092553