| L(s) = 1 | + 0.801·3-s − 3.55·5-s + 0.142·7-s − 2.35·9-s − 5.09·11-s − 3.44·13-s − 2.84·15-s + 5.28·17-s − 4.97·19-s + 0.113·21-s − 1.58·23-s + 7.62·25-s − 4.29·27-s + 5.53·29-s + 0.382·31-s − 4.08·33-s − 0.504·35-s − 6.21·37-s − 2.76·39-s − 11.4·41-s + 9.71·43-s + 8.37·45-s + 9.64·47-s − 6.97·49-s + 4.23·51-s − 10.7·53-s + 18.1·55-s + ⋯ |
| L(s) = 1 | + 0.462·3-s − 1.58·5-s + 0.0536·7-s − 0.786·9-s − 1.53·11-s − 0.955·13-s − 0.734·15-s + 1.28·17-s − 1.14·19-s + 0.0248·21-s − 0.329·23-s + 1.52·25-s − 0.826·27-s + 1.02·29-s + 0.0686·31-s − 0.710·33-s − 0.0852·35-s − 1.02·37-s − 0.442·39-s − 1.78·41-s + 1.48·43-s + 1.24·45-s + 1.40·47-s − 0.997·49-s + 0.593·51-s − 1.47·53-s + 2.44·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6330073350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6330073350\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 269 | \( 1 - T \) |
| good | 3 | \( 1 - 0.801T + 3T^{2} \) |
| 5 | \( 1 + 3.55T + 5T^{2} \) |
| 7 | \( 1 - 0.142T + 7T^{2} \) |
| 11 | \( 1 + 5.09T + 11T^{2} \) |
| 13 | \( 1 + 3.44T + 13T^{2} \) |
| 17 | \( 1 - 5.28T + 17T^{2} \) |
| 19 | \( 1 + 4.97T + 19T^{2} \) |
| 23 | \( 1 + 1.58T + 23T^{2} \) |
| 29 | \( 1 - 5.53T + 29T^{2} \) |
| 31 | \( 1 - 0.382T + 31T^{2} \) |
| 37 | \( 1 + 6.21T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 - 9.71T + 43T^{2} \) |
| 47 | \( 1 - 9.64T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 + 2.10T + 59T^{2} \) |
| 61 | \( 1 - 8.92T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 - 7.78T + 71T^{2} \) |
| 73 | \( 1 - 8.61T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 + 5.61T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 - 8.41T + 97T^{2} \) |
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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
−8.296129842860430483279152844179, −7.72294591809693113764955626836, −7.32367313503145051316664458925, −6.20671182338768656527290791269, −5.20252625967378560915463979993, −4.67723594421279885402725194247, −3.62867725077883429654692709453, −3.03333700248008130018470830300, −2.22762064301727296837879733045, −0.40821075168406269014565020978,
0.40821075168406269014565020978, 2.22762064301727296837879733045, 3.03333700248008130018470830300, 3.62867725077883429654692709453, 4.67723594421279885402725194247, 5.20252625967378560915463979993, 6.20671182338768656527290791269, 7.32367313503145051316664458925, 7.72294591809693113764955626836, 8.296129842860430483279152844179
