| L(s) = 1 | + 1.25·3-s − 3.52·5-s − 1.42·9-s + 11-s − 2.26·13-s − 4.42·15-s + 4.78·17-s − 2.51·19-s + 8.42·23-s + 7.42·25-s − 5.55·27-s − 6·29-s + 1.25·31-s + 1.25·33-s + 4.42·37-s − 2.84·39-s + 9.31·41-s + 10.8·43-s + 5.02·45-s − 4.78·47-s + 5.99·51-s + 8.84·53-s − 3.52·55-s − 3.15·57-s − 8.30·59-s − 0.240·61-s + 8·65-s + ⋯ |
| L(s) = 1 | + 0.724·3-s − 1.57·5-s − 0.474·9-s + 0.301·11-s − 0.629·13-s − 1.14·15-s + 1.15·17-s − 0.575·19-s + 1.75·23-s + 1.48·25-s − 1.06·27-s − 1.11·29-s + 0.225·31-s + 0.218·33-s + 0.727·37-s − 0.456·39-s + 1.45·41-s + 1.65·43-s + 0.748·45-s − 0.697·47-s + 0.840·51-s + 1.21·53-s − 0.475·55-s − 0.417·57-s − 1.08·59-s − 0.0308·61-s + 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.474337976\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.474337976\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 1.25T + 3T^{2} \) |
| 5 | \( 1 + 3.52T + 5T^{2} \) |
| 13 | \( 1 + 2.26T + 13T^{2} \) |
| 17 | \( 1 - 4.78T + 17T^{2} \) |
| 19 | \( 1 + 2.51T + 19T^{2} \) |
| 23 | \( 1 - 8.42T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 1.25T + 31T^{2} \) |
| 37 | \( 1 - 4.42T + 37T^{2} \) |
| 41 | \( 1 - 9.31T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 + 4.78T + 47T^{2} \) |
| 53 | \( 1 - 8.84T + 53T^{2} \) |
| 59 | \( 1 + 8.30T + 59T^{2} \) |
| 61 | \( 1 + 0.240T + 61T^{2} \) |
| 67 | \( 1 - 3.57T + 67T^{2} \) |
| 71 | \( 1 + 4.42T + 71T^{2} \) |
| 73 | \( 1 - 14.3T + 73T^{2} \) |
| 79 | \( 1 - 6T + 79T^{2} \) |
| 83 | \( 1 - 9.56T + 83T^{2} \) |
| 89 | \( 1 - 8.54T + 89T^{2} \) |
| 97 | \( 1 - 1.01T + 97T^{2} \) |
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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
−9.058242811752511796180438861131, −8.150593094917937008034781320610, −7.66998204858068169415901088039, −7.10528542978807920699284199772, −5.91151454529409096277560990414, −4.90005703289788419224181554012, −3.98675530429694916983217319469, −3.31874128829075414022403893943, −2.48304971080124585014227791748, −0.75874624783166166652060892908,
0.75874624783166166652060892908, 2.48304971080124585014227791748, 3.31874128829075414022403893943, 3.98675530429694916983217319469, 4.90005703289788419224181554012, 5.91151454529409096277560990414, 7.10528542978807920699284199772, 7.66998204858068169415901088039, 8.150593094917937008034781320610, 9.058242811752511796180438861131
