L(s) = 1 | − 3.00·3-s + 2.62·5-s − 7-s + 6.03·9-s + 11-s − 13-s − 7.88·15-s − 0.640·17-s − 5.26·19-s + 3.00·21-s + 3.73·23-s + 1.87·25-s − 9.13·27-s − 2.10·29-s − 10.8·31-s − 3.00·33-s − 2.62·35-s − 5.89·37-s + 3.00·39-s + 10.7·41-s + 2.17·43-s + 15.8·45-s + 3.39·47-s + 49-s + 1.92·51-s + 3.38·53-s + 2.62·55-s + ⋯ |
L(s) = 1 | − 1.73·3-s + 1.17·5-s − 0.377·7-s + 2.01·9-s + 0.301·11-s − 0.277·13-s − 2.03·15-s − 0.155·17-s − 1.20·19-s + 0.656·21-s + 0.779·23-s + 0.374·25-s − 1.75·27-s − 0.391·29-s − 1.94·31-s − 0.523·33-s − 0.443·35-s − 0.969·37-s + 0.481·39-s + 1.67·41-s + 0.332·43-s + 2.36·45-s + 0.495·47-s + 0.142·49-s + 0.269·51-s + 0.465·53-s + 0.353·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.005678609\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.005678609\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 + 3.00T + 3T^{2} \) |
| 5 | \( 1 - 2.62T + 5T^{2} \) |
| 17 | \( 1 + 0.640T + 17T^{2} \) |
| 19 | \( 1 + 5.26T + 19T^{2} \) |
| 23 | \( 1 - 3.73T + 23T^{2} \) |
| 29 | \( 1 + 2.10T + 29T^{2} \) |
| 31 | \( 1 + 10.8T + 31T^{2} \) |
| 37 | \( 1 + 5.89T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 - 2.17T + 43T^{2} \) |
| 47 | \( 1 - 3.39T + 47T^{2} \) |
| 53 | \( 1 - 3.38T + 53T^{2} \) |
| 59 | \( 1 + 3.21T + 59T^{2} \) |
| 61 | \( 1 - 14.8T + 61T^{2} \) |
| 67 | \( 1 - 7.51T + 67T^{2} \) |
| 71 | \( 1 + 2.60T + 71T^{2} \) |
| 73 | \( 1 + 0.811T + 73T^{2} \) |
| 79 | \( 1 - 9.43T + 79T^{2} \) |
| 83 | \( 1 + 16.0T + 83T^{2} \) |
| 89 | \( 1 + 8.41T + 89T^{2} \) |
| 97 | \( 1 - 8.56T + 97T^{2} \) |
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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
−7.44371738251273376831023283453, −6.89088426969203826111003670674, −6.35247355800254782380362709808, −5.63002990947456119033844153012, −5.42662721491819525374714149262, −4.48122190958619329935750489971, −3.75680556402508006502116970654, −2.38959730801423049334865395645, −1.64102078710959124031392224990, −0.54862557925401252143866036642,
0.54862557925401252143866036642, 1.64102078710959124031392224990, 2.38959730801423049334865395645, 3.75680556402508006502116970654, 4.48122190958619329935750489971, 5.42662721491819525374714149262, 5.63002990947456119033844153012, 6.35247355800254782380362709808, 6.89088426969203826111003670674, 7.44371738251273376831023283453