| L(s) = 1 | − 5.93·3-s − 3.12·7-s + 8.18·9-s − 14.3·11-s + 66.6·13-s + 109.·17-s + 40.4·19-s + 18.5·21-s + 23·23-s + 111.·27-s − 122.·29-s + 172.·31-s + 84.9·33-s − 1.84·37-s − 395.·39-s − 220.·41-s − 48.9·43-s + 21.5·47-s − 333.·49-s − 648.·51-s + 15.7·53-s − 239.·57-s + 684.·59-s − 480.·61-s − 25.5·63-s + 171.·67-s − 136.·69-s + ⋯ |
| L(s) = 1 | − 1.14·3-s − 0.168·7-s + 0.303·9-s − 0.392·11-s + 1.42·13-s + 1.55·17-s + 0.488·19-s + 0.192·21-s + 0.208·23-s + 0.795·27-s − 0.787·29-s + 0.998·31-s + 0.447·33-s − 0.00819·37-s − 1.62·39-s − 0.840·41-s − 0.173·43-s + 0.0670·47-s − 0.971·49-s − 1.77·51-s + 0.0409·53-s − 0.557·57-s + 1.50·59-s − 1.00·61-s − 0.0510·63-s + 0.312·67-s − 0.238·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.414624926\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.414624926\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 + 5.93T + 27T^{2} \) |
| 7 | \( 1 + 3.12T + 343T^{2} \) |
| 11 | \( 1 + 14.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 66.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 109.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 40.4T + 6.85e3T^{2} \) |
| 29 | \( 1 + 122.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 172.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 1.84T + 5.06e4T^{2} \) |
| 41 | \( 1 + 220.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 48.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 21.5T + 1.03e5T^{2} \) |
| 53 | \( 1 - 15.7T + 1.48e5T^{2} \) |
| 59 | \( 1 - 684.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 480.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 171.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 382.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 381.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 345.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 243.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 618.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.38e3T + 9.12e5T^{2} \) |
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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
−8.540773684933173430963264730116, −7.915847048814705137225141866641, −6.93191894708555962287123444305, −6.15811168936106854364549665657, −5.57916764059452735513713186955, −4.94725535701256836225978295373, −3.75411703370252655648122576070, −2.98770944395996472148811237393, −1.44385001747797443645516318760, −0.61565293223525701777499526321,
0.61565293223525701777499526321, 1.44385001747797443645516318760, 2.98770944395996472148811237393, 3.75411703370252655648122576070, 4.94725535701256836225978295373, 5.57916764059452735513713186955, 6.15811168936106854364549665657, 6.93191894708555962287123444305, 7.915847048814705137225141866641, 8.540773684933173430963264730116
