| L(s) = 1 | − 0.477·2-s − 0.323·3-s − 1.77·4-s + 0.154·6-s − 2.68·7-s + 1.80·8-s − 2.89·9-s + 0.572·12-s + 4.66·13-s + 1.28·14-s + 2.68·16-s + 4.62·17-s + 1.38·18-s − 4.34·19-s + 0.867·21-s − 2.77·23-s − 0.581·24-s − 2.22·26-s + 1.90·27-s + 4.75·28-s − 3.01·29-s + 2.38·31-s − 4.88·32-s − 2.20·34-s + 5.13·36-s + 10.6·37-s + 2.07·38-s + ⋯ |
| L(s) = 1 | − 0.337·2-s − 0.186·3-s − 0.886·4-s + 0.0629·6-s − 1.01·7-s + 0.636·8-s − 0.965·9-s + 0.165·12-s + 1.29·13-s + 0.342·14-s + 0.671·16-s + 1.12·17-s + 0.325·18-s − 0.995·19-s + 0.189·21-s − 0.578·23-s − 0.118·24-s − 0.436·26-s + 0.366·27-s + 0.899·28-s − 0.559·29-s + 0.428·31-s − 0.863·32-s − 0.378·34-s + 0.855·36-s + 1.74·37-s + 0.336·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.477T + 2T^{2} \) |
| 3 | \( 1 + 0.323T + 3T^{2} \) |
| 7 | \( 1 + 2.68T + 7T^{2} \) |
| 13 | \( 1 - 4.66T + 13T^{2} \) |
| 17 | \( 1 - 4.62T + 17T^{2} \) |
| 19 | \( 1 + 4.34T + 19T^{2} \) |
| 23 | \( 1 + 2.77T + 23T^{2} \) |
| 29 | \( 1 + 3.01T + 29T^{2} \) |
| 31 | \( 1 - 2.38T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 + 2.21T + 41T^{2} \) |
| 43 | \( 1 - 7.06T + 43T^{2} \) |
| 47 | \( 1 + 4.36T + 47T^{2} \) |
| 53 | \( 1 - 6.33T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 + 3.98T + 61T^{2} \) |
| 67 | \( 1 + 7.31T + 67T^{2} \) |
| 71 | \( 1 - 1.19T + 71T^{2} \) |
| 73 | \( 1 + 1.02T + 73T^{2} \) |
| 79 | \( 1 - 3.50T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 - 2.76T + 89T^{2} \) |
| 97 | \( 1 + 18.5T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.325927376477855969526861204623, −7.953942018595726928240785979831, −6.73106514945735499131701135406, −5.93058801268673502156611849330, −5.53308939571691545605495331820, −4.26013875604725393084786097850, −3.64554049099246509150483798809, −2.70967446493084728628464806604, −1.13925083867890145318362143588, 0,
1.13925083867890145318362143588, 2.70967446493084728628464806604, 3.64554049099246509150483798809, 4.26013875604725393084786097850, 5.53308939571691545605495331820, 5.93058801268673502156611849330, 6.73106514945735499131701135406, 7.953942018595726928240785979831, 8.325927376477855969526861204623
