| L(s) = 1 | + 0.368·3-s − 1.72·5-s − 0.634·7-s − 2.86·9-s − 6.09·11-s + 6.49·13-s − 0.634·15-s − 2.10·17-s + 4.13·19-s − 0.233·21-s − 2.03·25-s − 2.16·27-s + 0.137·29-s + 6.17·31-s − 2.24·33-s + 1.09·35-s − 9.22·37-s + 2.39·39-s − 3.58·41-s − 5.59·43-s + 4.93·45-s + 7.61·47-s − 6.59·49-s − 0.776·51-s + 7.23·53-s + 10.5·55-s + 1.52·57-s + ⋯ |
| L(s) = 1 | + 0.212·3-s − 0.770·5-s − 0.239·7-s − 0.954·9-s − 1.83·11-s + 1.80·13-s − 0.163·15-s − 0.511·17-s + 0.948·19-s − 0.0510·21-s − 0.407·25-s − 0.415·27-s + 0.0255·29-s + 1.10·31-s − 0.391·33-s + 0.184·35-s − 1.51·37-s + 0.383·39-s − 0.560·41-s − 0.853·43-s + 0.735·45-s + 1.11·47-s − 0.942·49-s − 0.108·51-s + 0.993·53-s + 1.41·55-s + 0.201·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.094942672\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.094942672\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 0.368T + 3T^{2} \) |
| 5 | \( 1 + 1.72T + 5T^{2} \) |
| 7 | \( 1 + 0.634T + 7T^{2} \) |
| 11 | \( 1 + 6.09T + 11T^{2} \) |
| 13 | \( 1 - 6.49T + 13T^{2} \) |
| 17 | \( 1 + 2.10T + 17T^{2} \) |
| 19 | \( 1 - 4.13T + 19T^{2} \) |
| 29 | \( 1 - 0.137T + 29T^{2} \) |
| 31 | \( 1 - 6.17T + 31T^{2} \) |
| 37 | \( 1 + 9.22T + 37T^{2} \) |
| 41 | \( 1 + 3.58T + 41T^{2} \) |
| 43 | \( 1 + 5.59T + 43T^{2} \) |
| 47 | \( 1 - 7.61T + 47T^{2} \) |
| 53 | \( 1 - 7.23T + 53T^{2} \) |
| 59 | \( 1 + 9.94T + 59T^{2} \) |
| 61 | \( 1 - 1.63T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 - 4.13T + 71T^{2} \) |
| 73 | \( 1 - 4.17T + 73T^{2} \) |
| 79 | \( 1 + 2.88T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 + 1.91T + 89T^{2} \) |
| 97 | \( 1 - 6.18T + 97T^{2} \) |
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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
−8.408914034173368810064912469724, −7.85952329414389409489354798192, −7.05322339895924539142205223817, −6.09603691358626962160387361189, −5.48306889152778039457531675173, −4.68291303913470349946583269380, −3.50729262088982170058017086621, −3.18189445203539948293191646204, −2.09010490501177655053008203858, −0.56132019561071802782895764419,
0.56132019561071802782895764419, 2.09010490501177655053008203858, 3.18189445203539948293191646204, 3.50729262088982170058017086621, 4.68291303913470349946583269380, 5.48306889152778039457531675173, 6.09603691358626962160387361189, 7.05322339895924539142205223817, 7.85952329414389409489354798192, 8.408914034173368810064912469724
