| L(s) = 1 | + 5-s − 1.14·7-s − 5.89·11-s − 4.89·13-s + 5.89·17-s − 2.34·19-s − 23-s + 25-s − 3.74·29-s + 5.68·31-s − 1.14·35-s + 4·37-s + 1.05·41-s + 11.4·43-s − 7.74·47-s − 5.68·49-s − 12.9·53-s − 5.89·55-s − 0.797·59-s + 13.8·61-s − 4.89·65-s − 15.5·67-s − 2.94·71-s + 6.32·73-s + 6.74·77-s + 0.912·83-s + 5.89·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.432·7-s − 1.77·11-s − 1.35·13-s + 1.42·17-s − 0.538·19-s − 0.208·23-s + 0.200·25-s − 0.695·29-s + 1.02·31-s − 0.193·35-s + 0.657·37-s + 0.165·41-s + 1.75·43-s − 1.12·47-s − 0.812·49-s − 1.78·53-s − 0.794·55-s − 0.103·59-s + 1.77·61-s − 0.606·65-s − 1.90·67-s − 0.349·71-s + 0.740·73-s + 0.768·77-s + 0.100·83-s + 0.639·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.298944815\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.298944815\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 1.14T + 7T^{2} \) |
| 11 | \( 1 + 5.89T + 11T^{2} \) |
| 13 | \( 1 + 4.89T + 13T^{2} \) |
| 17 | \( 1 - 5.89T + 17T^{2} \) |
| 19 | \( 1 + 2.34T + 19T^{2} \) |
| 29 | \( 1 + 3.74T + 29T^{2} \) |
| 31 | \( 1 - 5.68T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 1.05T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + 7.74T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 + 0.797T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 + 15.5T + 67T^{2} \) |
| 71 | \( 1 + 2.94T + 71T^{2} \) |
| 73 | \( 1 - 6.32T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 0.912T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
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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
−7.80736028605217387281971356928, −7.28124524388511297371163756833, −6.31879384328095965349308238214, −5.70326090631985303720864176355, −5.06758504456611364617480792213, −4.46282235072243650301698966265, −3.22922345547180989544071803473, −2.71102806546761102333311341425, −1.93608396133256169154574182960, −0.53136143387949981017206889952,
0.53136143387949981017206889952, 1.93608396133256169154574182960, 2.71102806546761102333311341425, 3.22922345547180989544071803473, 4.46282235072243650301698966265, 5.06758504456611364617480792213, 5.70326090631985303720864176355, 6.31879384328095965349308238214, 7.28124524388511297371163756833, 7.80736028605217387281971356928
