| L(s) = 1 | + 25·5-s + 99.3·7-s − 377.·11-s − 388.·13-s + 1.15e3·17-s + 246.·19-s − 195.·23-s + 625·25-s − 3.26e3·29-s + 160.·31-s + 2.48e3·35-s − 9.12e3·37-s + 1.23e4·41-s + 8.30e3·43-s − 1.10e4·47-s − 6.93e3·49-s + 6.02e3·53-s − 9.42e3·55-s + 1.55e3·59-s − 5.09e4·61-s − 9.70e3·65-s + 1.55e4·67-s − 5.09e4·71-s − 2.52e4·73-s − 3.74e4·77-s + 1.68e4·79-s − 1.08e5·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.766·7-s − 0.939·11-s − 0.637·13-s + 0.972·17-s + 0.156·19-s − 0.0771·23-s + 0.200·25-s − 0.721·29-s + 0.0300·31-s + 0.342·35-s − 1.09·37-s + 1.15·41-s + 0.684·43-s − 0.727·47-s − 0.412·49-s + 0.294·53-s − 0.420·55-s + 0.0582·59-s − 1.75·61-s − 0.284·65-s + 0.422·67-s − 1.19·71-s − 0.553·73-s − 0.720·77-s + 0.304·79-s − 1.73·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 25T \) |
| good | 7 | \( 1 - 99.3T + 1.68e4T^{2} \) |
| 11 | \( 1 + 377.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 388.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.15e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 246.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 195.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.26e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 160.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.12e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.23e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.30e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.10e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 6.02e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.55e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.09e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.55e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.09e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.52e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.68e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.08e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.61e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.78e5T + 8.58e9T^{2} \) |
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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
−8.725906140221332040986928523975, −7.77604711793657610338807703202, −7.32665200463068951560539358136, −6.01036640797103981253828954633, −5.30305351935533750835968894792, −4.56750394621856169310124165835, −3.26374607725803957656055880919, −2.27855750996644739347962331952, −1.31158620239993245865347300358, 0,
1.31158620239993245865347300358, 2.27855750996644739347962331952, 3.26374607725803957656055880919, 4.56750394621856169310124165835, 5.30305351935533750835968894792, 6.01036640797103981253828954633, 7.32665200463068951560539358136, 7.77604711793657610338807703202, 8.725906140221332040986928523975
