| L(s) = 1 | − 3.38·3-s − 1.60·5-s + 8.42·9-s + 11-s − 4.98·13-s + 5.42·15-s − 1.77·17-s + 6.76·19-s − 1.42·23-s − 2.42·25-s − 18.3·27-s − 6·29-s − 3.38·31-s − 3.38·33-s − 5.42·37-s + 16.8·39-s + 8.19·41-s − 8.84·43-s − 13.5·45-s + 1.77·47-s + 6·51-s − 10.8·53-s − 1.60·55-s − 22.8·57-s + 0.170·59-s + 11.7·61-s + 8.00·65-s + ⋯ |
| L(s) = 1 | − 1.95·3-s − 0.717·5-s + 2.80·9-s + 0.301·11-s − 1.38·13-s + 1.40·15-s − 0.430·17-s + 1.55·19-s − 0.297·23-s − 0.484·25-s − 3.52·27-s − 1.11·29-s − 0.607·31-s − 0.588·33-s − 0.891·37-s + 2.69·39-s + 1.27·41-s − 1.34·43-s − 2.01·45-s + 0.258·47-s + 0.840·51-s − 1.49·53-s − 0.216·55-s − 3.02·57-s + 0.0221·59-s + 1.50·61-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
| \(L(1)\) |
\(\approx\) |
\(0.4506324082\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4506324082\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.38T + 3T^{2} \) |
| 5 | \( 1 + 1.60T + 5T^{2} \) |
| 13 | \( 1 + 4.98T + 13T^{2} \) |
| 17 | \( 1 + 1.77T + 17T^{2} \) |
| 19 | \( 1 - 6.76T + 19T^{2} \) |
| 23 | \( 1 + 1.42T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 3.38T + 31T^{2} \) |
| 37 | \( 1 + 5.42T + 37T^{2} \) |
| 41 | \( 1 - 8.19T + 41T^{2} \) |
| 43 | \( 1 + 8.84T + 43T^{2} \) |
| 47 | \( 1 - 1.77T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 - 0.170T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 - 5.42T + 71T^{2} \) |
| 73 | \( 1 + 5.32T + 73T^{2} \) |
| 79 | \( 1 - 6T + 79T^{2} \) |
| 83 | \( 1 + 3.55T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 - 8.36T + 97T^{2} \) |
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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
−9.549664118078488381676418062092, −7.982175253107107225388701021548, −7.23726836107023057620037409263, −6.84773913963929126381642775461, −5.72869878991340770530367782862, −5.21504927066829556199417866522, −4.43971807031094515002508953222, −3.58203322199081201485166777753, −1.84316992869418754902218806975, −0.48101441231286537073723328461,
0.48101441231286537073723328461, 1.84316992869418754902218806975, 3.58203322199081201485166777753, 4.43971807031094515002508953222, 5.21504927066829556199417866522, 5.72869878991340770530367782862, 6.84773913963929126381642775461, 7.23726836107023057620037409263, 7.982175253107107225388701021548, 9.549664118078488381676418062092
