L(s) = 1 | + 3.11·3-s + (0.660 − 2.13i)5-s − i·7-s + 6.70·9-s + 0.780i·11-s + 3.09·13-s + (2.05 − 6.65i)15-s + 7.31i·17-s + 0.207i·19-s − 3.11i·21-s − 1.52i·23-s + (−4.12 − 2.82i)25-s + 11.5·27-s − 3.60i·29-s + 7.94·31-s + ⋯ |
L(s) = 1 | + 1.79·3-s + (0.295 − 0.955i)5-s − 0.377i·7-s + 2.23·9-s + 0.235i·11-s + 0.857·13-s + (0.531 − 1.71i)15-s + 1.77i·17-s + 0.0476i·19-s − 0.679i·21-s − 0.318i·23-s + (−0.825 − 0.564i)25-s + 2.22·27-s − 0.669i·29-s + 1.42·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 + 0.532i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.846 + 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.887061939\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.887061939\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.660 + 2.13i)T \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 - 3.11T + 3T^{2} \) |
| 11 | \( 1 - 0.780iT - 11T^{2} \) |
| 13 | \( 1 - 3.09T + 13T^{2} \) |
| 17 | \( 1 - 7.31iT - 17T^{2} \) |
| 19 | \( 1 - 0.207iT - 19T^{2} \) |
| 23 | \( 1 + 1.52iT - 23T^{2} \) |
| 29 | \( 1 + 3.60iT - 29T^{2} \) |
| 31 | \( 1 - 7.94T + 31T^{2} \) |
| 37 | \( 1 + 3.70T + 37T^{2} \) |
| 41 | \( 1 - 6.24T + 41T^{2} \) |
| 43 | \( 1 + 6.61T + 43T^{2} \) |
| 47 | \( 1 - 2.61iT - 47T^{2} \) |
| 53 | \( 1 + 8.69T + 53T^{2} \) |
| 59 | \( 1 + 14.6iT - 59T^{2} \) |
| 61 | \( 1 - 6.37iT - 61T^{2} \) |
| 67 | \( 1 - 7.21T + 67T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 + 0.502iT - 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 + 16.3T + 89T^{2} \) |
| 97 | \( 1 - 10.2iT - 97T^{2} \) |
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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
−8.744787979552181302181167681761, −8.271758936354635953026626331539, −7.905195226701827634911491777083, −6.73400199963792186620814346160, −5.91980251084740537146978105914, −4.55424214477906679281470774823, −4.05136710740080593503231902381, −3.18134508266922037776049310162, −2.01148871658127960030773185239, −1.28726295993871255168934211146,
1.49071725599085498100560542481, 2.66979244037893389263184810228, 3.00368961928087653518417397466, 3.88203062330182885000474332296, 5.00039805637513882826133241379, 6.18916926642811938535547093109, 7.02141288166103109656568920161, 7.59994504783538767804611060655, 8.447675445205988942092677566248, 9.036468239994979284064396061063