L(s) = 1 | + 4·7-s − 3·9-s + 11-s − 6·13-s + 6·17-s − 4·19-s + 4·23-s − 2·29-s − 8·31-s + 10·37-s + 10·41-s + 4·47-s + 9·49-s + 10·53-s + 4·59-s − 2·61-s − 12·63-s − 8·67-s + 14·73-s + 4·77-s + 16·79-s + 9·81-s − 8·83-s − 6·89-s − 24·91-s − 2·97-s − 3·99-s + ⋯ |
L(s) = 1 | + 1.51·7-s − 9-s + 0.301·11-s − 1.66·13-s + 1.45·17-s − 0.917·19-s + 0.834·23-s − 0.371·29-s − 1.43·31-s + 1.64·37-s + 1.56·41-s + 0.583·47-s + 9/7·49-s + 1.37·53-s + 0.520·59-s − 0.256·61-s − 1.51·63-s − 0.977·67-s + 1.63·73-s + 0.455·77-s + 1.80·79-s + 81-s − 0.878·83-s − 0.635·89-s − 2.51·91-s − 0.203·97-s − 0.301·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.027000377\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.027000377\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.238867245718623313561169683063, −7.66073218752774493934165713546, −7.18329864658559389687378602652, −5.94194779788411527449061590692, −5.37462190663999254068940531745, −4.74959653924644122740976703463, −3.89465729413229450468011084000, −2.72560030668527945539611648478, −2.05226332576611798463637696772, −0.802746013531920844432735139795,
0.802746013531920844432735139795, 2.05226332576611798463637696772, 2.72560030668527945539611648478, 3.89465729413229450468011084000, 4.74959653924644122740976703463, 5.37462190663999254068940531745, 5.94194779788411527449061590692, 7.18329864658559389687378602652, 7.66073218752774493934165713546, 8.238867245718623313561169683063