| L(s) = 1 | − 2·7-s − 4·11-s − 13-s − 6·17-s + 6·19-s + 2·29-s − 6·31-s − 10·37-s − 8·41-s − 12·43-s + 12·47-s − 3·49-s − 6·53-s + 2·61-s − 2·67-s + 8·71-s − 14·73-s + 8·77-s + 4·79-s + 8·83-s − 4·89-s + 2·91-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 1.20·11-s − 0.277·13-s − 1.45·17-s + 1.37·19-s + 0.371·29-s − 1.07·31-s − 1.64·37-s − 1.24·41-s − 1.82·43-s + 1.75·47-s − 3/7·49-s − 0.824·53-s + 0.256·61-s − 0.244·67-s + 0.949·71-s − 1.63·73-s + 0.911·77-s + 0.450·79-s + 0.878·83-s − 0.423·89-s + 0.209·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.15702280566539, −13.70021268750372, −13.34444967426045, −12.96463447466038, −12.34187491817853, −11.94335919449543, −11.36534426344964, −10.77516624567909, −10.37740487371721, −9.889459331301458, −9.387082435807941, −8.831061030871622, −8.381513254606914, −7.745639792774034, −7.125248913392063, −6.861983637560176, −6.218726596113574, −5.486369203423962, −5.097009681496116, −4.645680460203536, −3.676555959395790, −3.332007683155390, −2.659416266857012, −2.078055609838142, −1.308707189996465, 0, 0,
1.308707189996465, 2.078055609838142, 2.659416266857012, 3.332007683155390, 3.676555959395790, 4.645680460203536, 5.097009681496116, 5.486369203423962, 6.218726596113574, 6.861983637560176, 7.125248913392063, 7.745639792774034, 8.381513254606914, 8.831061030871622, 9.387082435807941, 9.889459331301458, 10.37740487371721, 10.77516624567909, 11.36534426344964, 11.94335919449543, 12.34187491817853, 12.96463447466038, 13.34444967426045, 13.70021268750372, 14.15702280566539
