L(s) = 1 | + 5-s − 2.26·7-s − 0.162·11-s + 7.01·13-s + 3.86·17-s + 1.83·19-s + 23-s + 25-s + 7.93·29-s − 5.15·31-s − 2.26·35-s + 6.51·37-s − 1.28·41-s − 5.17·43-s − 4.41·47-s − 1.88·49-s + 0.792·53-s − 0.162·55-s − 4.75·59-s + 13.0·61-s + 7.01·65-s − 1.99·67-s + 11.9·71-s − 11.9·73-s + 0.368·77-s + 9.21·79-s + 7.90·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.854·7-s − 0.0491·11-s + 1.94·13-s + 0.936·17-s + 0.421·19-s + 0.208·23-s + 0.200·25-s + 1.47·29-s − 0.925·31-s − 0.382·35-s + 1.07·37-s − 0.200·41-s − 0.789·43-s − 0.644·47-s − 0.269·49-s + 0.108·53-s − 0.0219·55-s − 0.618·59-s + 1.67·61-s + 0.870·65-s − 0.243·67-s + 1.41·71-s − 1.39·73-s + 0.0419·77-s + 1.03·79-s + 0.867·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.433098195\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.433098195\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 + 2.26T + 7T^{2} \) |
| 11 | \( 1 + 0.162T + 11T^{2} \) |
| 13 | \( 1 - 7.01T + 13T^{2} \) |
| 17 | \( 1 - 3.86T + 17T^{2} \) |
| 19 | \( 1 - 1.83T + 19T^{2} \) |
| 29 | \( 1 - 7.93T + 29T^{2} \) |
| 31 | \( 1 + 5.15T + 31T^{2} \) |
| 37 | \( 1 - 6.51T + 37T^{2} \) |
| 41 | \( 1 + 1.28T + 41T^{2} \) |
| 43 | \( 1 + 5.17T + 43T^{2} \) |
| 47 | \( 1 + 4.41T + 47T^{2} \) |
| 53 | \( 1 - 0.792T + 53T^{2} \) |
| 59 | \( 1 + 4.75T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 + 1.99T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 - 9.21T + 79T^{2} \) |
| 83 | \( 1 - 7.90T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 - 0.560T + 97T^{2} \) |
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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
−7.987597150546807808082413469661, −6.86899219736573873617212727604, −6.46441871027320960374710462849, −5.78898508095145441289812759005, −5.20359496888421860730403541818, −4.12573297826027897393181891864, −3.39754294267740865089801568863, −2.87071965778548710600295564363, −1.61607583415162811502754656433, −0.817989701220327307007684618086,
0.817989701220327307007684618086, 1.61607583415162811502754656433, 2.87071965778548710600295564363, 3.39754294267740865089801568863, 4.12573297826027897393181891864, 5.20359496888421860730403541818, 5.78898508095145441289812759005, 6.46441871027320960374710462849, 6.86899219736573873617212727604, 7.987597150546807808082413469661