L(s) = 1 | − 0.463·3-s − 1.98·5-s − 7-s − 2.78·9-s − 0.628·11-s − 3.19·13-s + 0.919·15-s − 4.85·17-s − 0.161·19-s + 0.463·21-s − 3.20·23-s − 1.06·25-s + 2.68·27-s − 1.41·29-s + 5.34·31-s + 0.291·33-s + 1.98·35-s − 2.87·37-s + 1.47·39-s − 9.57·41-s − 9.71·43-s + 5.52·45-s − 9.70·47-s + 49-s + 2.25·51-s + 10.8·53-s + 1.24·55-s + ⋯ |
L(s) = 1 | − 0.267·3-s − 0.886·5-s − 0.377·7-s − 0.928·9-s − 0.189·11-s − 0.884·13-s + 0.237·15-s − 1.17·17-s − 0.0369·19-s + 0.101·21-s − 0.668·23-s − 0.213·25-s + 0.516·27-s − 0.261·29-s + 0.959·31-s + 0.0507·33-s + 0.335·35-s − 0.473·37-s + 0.236·39-s − 1.49·41-s − 1.48·43-s + 0.823·45-s − 1.41·47-s + 0.142·49-s + 0.315·51-s + 1.48·53-s + 0.167·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1773283696\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1773283696\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 + 0.463T + 3T^{2} \) |
| 5 | \( 1 + 1.98T + 5T^{2} \) |
| 11 | \( 1 + 0.628T + 11T^{2} \) |
| 13 | \( 1 + 3.19T + 13T^{2} \) |
| 17 | \( 1 + 4.85T + 17T^{2} \) |
| 19 | \( 1 + 0.161T + 19T^{2} \) |
| 23 | \( 1 + 3.20T + 23T^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 31 | \( 1 - 5.34T + 31T^{2} \) |
| 37 | \( 1 + 2.87T + 37T^{2} \) |
| 41 | \( 1 + 9.57T + 41T^{2} \) |
| 43 | \( 1 + 9.71T + 43T^{2} \) |
| 47 | \( 1 + 9.70T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 - 2.12T + 59T^{2} \) |
| 61 | \( 1 + 2.46T + 61T^{2} \) |
| 67 | \( 1 + 12.6T + 67T^{2} \) |
| 71 | \( 1 + 7.18T + 71T^{2} \) |
| 73 | \( 1 + 9.04T + 73T^{2} \) |
| 79 | \( 1 + 9.58T + 79T^{2} \) |
| 83 | \( 1 - 7.49T + 83T^{2} \) |
| 89 | \( 1 + 2.49T + 89T^{2} \) |
| 97 | \( 1 - 5.89T + 97T^{2} \) |
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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
−7.988364762622308125768945281875, −7.18642144821869974239413212938, −6.58601780013575187739794343805, −5.87106493705104129876012337550, −5.00454070258165636644892501240, −4.43716264954006294153191748344, −3.51110292163101067980372145263, −2.82399680702691540820203196860, −1.87969639710193875641024781970, −0.20195567754837795987658225791,
0.20195567754837795987658225791, 1.87969639710193875641024781970, 2.82399680702691540820203196860, 3.51110292163101067980372145263, 4.43716264954006294153191748344, 5.00454070258165636644892501240, 5.87106493705104129876012337550, 6.58601780013575187739794343805, 7.18642144821869974239413212938, 7.988364762622308125768945281875