L(s) = 1 | − 2.82·5-s − 4·7-s + 1.41·11-s − 2.82·13-s + 4·17-s + 7.07·19-s + 4·23-s + 3.00·25-s + 8.48·29-s − 8·31-s + 11.3·35-s − 2.82·37-s − 2·41-s − 4.24·43-s + 9·49-s + 2.82·53-s − 4.00·55-s + 4.24·59-s + 8.48·61-s + 8.00·65-s − 4.24·67-s − 4·71-s − 4·73-s − 5.65·77-s + 8·79-s − 9.89·83-s − 11.3·85-s + ⋯ |
L(s) = 1 | − 1.26·5-s − 1.51·7-s + 0.426·11-s − 0.784·13-s + 0.970·17-s + 1.62·19-s + 0.834·23-s + 0.600·25-s + 1.57·29-s − 1.43·31-s + 1.91·35-s − 0.464·37-s − 0.312·41-s − 0.646·43-s + 1.28·49-s + 0.388·53-s − 0.539·55-s + 0.552·59-s + 1.08·61-s + 0.992·65-s − 0.518·67-s − 0.474·71-s − 0.468·73-s − 0.644·77-s + 0.900·79-s − 1.08·83-s − 1.22·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 2.82T + 5T^{2} \) |
| 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 7.07T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 8.48T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 2.82T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4.24T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 2.82T + 53T^{2} \) |
| 59 | \( 1 - 4.24T + 59T^{2} \) |
| 61 | \( 1 - 8.48T + 61T^{2} \) |
| 67 | \( 1 + 4.24T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 9.89T + 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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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
−7.84903969390894661418510599698, −7.08776627010720449682293630352, −6.86157465868413329029464060919, −5.69959635188634512609660275395, −5.02077845689519862822651555274, −3.95580437302241635706835372986, −3.34370138412000004613672537636, −2.82253054695038920013746127298, −1.09888997650423189458245902236, 0,
1.09888997650423189458245902236, 2.82253054695038920013746127298, 3.34370138412000004613672537636, 3.95580437302241635706835372986, 5.02077845689519862822651555274, 5.69959635188634512609660275395, 6.86157465868413329029464060919, 7.08776627010720449682293630352, 7.84903969390894661418510599698