| 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
| \(L(1)\) |
\(\approx\) |
\(2.936571395\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.936571395\) |
| \(L(\frac{3}{2})\) |
|
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
−8.395806183162740858462524416838, −7.947081322537419518130696236391, −6.61334105686758533130677459728, −6.15158775431881716126368485955, −5.79016981843974477625117853800, −4.49427165626507687592778997130, −3.81384451025534629662437678645, −2.96382533862682798918990494120, −1.83113609877556110010025500643, −1.05447424687138421819906381000,
1.05447424687138421819906381000, 1.83113609877556110010025500643, 2.96382533862682798918990494120, 3.81384451025534629662437678645, 4.49427165626507687592778997130, 5.79016981843974477625117853800, 6.15158775431881716126368485955, 6.61334105686758533130677459728, 7.947081322537419518130696236391, 8.395806183162740858462524416838
