| L(s) = 1 | + 2.52·2-s + 3-s + 4.37·4-s + 2.52·6-s + 0.792·7-s + 5.98·8-s + 9-s + 4.37·12-s − 0.147·13-s + 2·14-s + 6.37·16-s + 6.63·17-s + 2.52·18-s − 4.40·19-s + 0.792·21-s − 8·23-s + 5.98·24-s − 0.372·26-s + 27-s + 3.46·28-s + 10.0·29-s + 2.37·31-s + 4.10·32-s + 16.7·34-s + 4.37·36-s − 5·37-s − 11.1·38-s + ⋯ |
| L(s) = 1 | + 1.78·2-s + 0.577·3-s + 2.18·4-s + 1.03·6-s + 0.299·7-s + 2.11·8-s + 0.333·9-s + 1.26·12-s − 0.0409·13-s + 0.534·14-s + 1.59·16-s + 1.60·17-s + 0.594·18-s − 1.01·19-s + 0.172·21-s − 1.66·23-s + 1.22·24-s − 0.0730·26-s + 0.192·27-s + 0.654·28-s + 1.87·29-s + 0.426·31-s + 0.726·32-s + 2.87·34-s + 0.728·36-s − 0.821·37-s − 1.80·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.254270318\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.254270318\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.52T + 2T^{2} \) |
| 7 | \( 1 - 0.792T + 7T^{2} \) |
| 13 | \( 1 + 0.147T + 13T^{2} \) |
| 17 | \( 1 - 6.63T + 17T^{2} \) |
| 19 | \( 1 + 4.40T + 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 - 10.0T + 29T^{2} \) |
| 31 | \( 1 - 2.37T + 31T^{2} \) |
| 37 | \( 1 + 5T + 37T^{2} \) |
| 41 | \( 1 - 6.92T + 41T^{2} \) |
| 43 | \( 1 - 9.94T + 43T^{2} \) |
| 47 | \( 1 - 8.74T + 47T^{2} \) |
| 53 | \( 1 - 1.25T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 5.98T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 5.19T + 73T^{2} \) |
| 79 | \( 1 + 6.78T + 79T^{2} \) |
| 83 | \( 1 - 8.51T + 83T^{2} \) |
| 89 | \( 1 - 5.48T + 89T^{2} \) |
| 97 | \( 1 - 9.37T + 97T^{2} \) |
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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
−7.72154438026617279659551729740, −6.76911046248198188467121624689, −6.22530610482231181570138402786, −5.54563662159613731142920332301, −4.88958027758379842975396179651, −4.03858824748523006369017109223, −3.77290766002613901238578097720, −2.69586312602104485176553011759, −2.29312644746199622747525219400, −1.16257966360915629108013288306,
1.16257966360915629108013288306, 2.29312644746199622747525219400, 2.69586312602104485176553011759, 3.77290766002613901238578097720, 4.03858824748523006369017109223, 4.88958027758379842975396179651, 5.54563662159613731142920332301, 6.22530610482231181570138402786, 6.76911046248198188467121624689, 7.72154438026617279659551729740
