L(s) = 1 | + 4·7-s + 2·11-s − 4·13-s − 17-s − 5·19-s − 5·23-s + 8·29-s + 7·31-s + 6·37-s + 6·41-s + 2·43-s − 8·47-s + 9·49-s − 9·53-s + 4·59-s + 13·61-s + 10·67-s − 6·71-s + 6·73-s + 8·77-s + 9·79-s + 17·83-s − 6·89-s − 16·91-s + 8·97-s + 12·101-s − 4·103-s + ⋯ |
L(s) = 1 | + 1.51·7-s + 0.603·11-s − 1.10·13-s − 0.242·17-s − 1.14·19-s − 1.04·23-s + 1.48·29-s + 1.25·31-s + 0.986·37-s + 0.937·41-s + 0.304·43-s − 1.16·47-s + 9/7·49-s − 1.23·53-s + 0.520·59-s + 1.66·61-s + 1.22·67-s − 0.712·71-s + 0.702·73-s + 0.911·77-s + 1.01·79-s + 1.86·83-s − 0.635·89-s − 1.67·91-s + 0.812·97-s + 1.19·101-s − 0.394·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.315278624\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.315278624\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.144163369382338898584731621486, −7.67390362300872939745052096435, −6.64865277490864503298061803625, −6.16418803174907872494596501811, −4.99420235073188230738456192430, −4.62832974992381239530366261114, −3.92618065486017069224476120854, −2.56319505257360539320180827920, −1.97997810687676281317793935761, −0.833779934765793390606047361505,
0.833779934765793390606047361505, 1.97997810687676281317793935761, 2.56319505257360539320180827920, 3.92618065486017069224476120854, 4.62832974992381239530366261114, 4.99420235073188230738456192430, 6.16418803174907872494596501811, 6.64865277490864503298061803625, 7.67390362300872939745052096435, 8.144163369382338898584731621486