L(s) = 1 | + (−0.294 − 7.99i)2-s + 15.5·3-s + (−63.8 + 4.71i)4-s + (71.1 − 102. i)5-s + (−4.59 − 124. i)6-s + 496.·7-s + (56.5 + 508. i)8-s + 243·9-s + (−842. − 538. i)10-s − 561. i·11-s + (−994. + 73.5i)12-s − 1.58e3i·13-s + (−146. − 3.96e3i)14-s + (1.10e3 − 1.60e3i)15-s + (4.05e3 − 601. i)16-s + 4.50e3i·17-s + ⋯ |
L(s) = 1 | + (−0.0368 − 0.999i)2-s + 0.577·3-s + (−0.997 + 0.0736i)4-s + (0.569 − 0.822i)5-s + (−0.0212 − 0.576i)6-s + 1.44·7-s + (0.110 + 0.993i)8-s + 0.333·9-s + (−0.842 − 0.538i)10-s − 0.421i·11-s + (−0.575 + 0.0425i)12-s − 0.721i·13-s + (−0.0533 − 1.44i)14-s + (0.328 − 0.474i)15-s + (0.989 − 0.146i)16-s + 0.916i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.506 + 0.862i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.506 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(1.17369 - 2.05168i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.17369 - 2.05168i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.294 + 7.99i)T \) |
| 3 | \( 1 - 15.5T \) |
| 5 | \( 1 + (-71.1 + 102. i)T \) |
good | 7 | \( 1 - 496.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 561. iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 1.58e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 4.50e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 1.01e4iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 1.61e4T + 1.48e8T^{2} \) |
| 29 | \( 1 - 2.00e3T + 5.94e8T^{2} \) |
| 31 | \( 1 + 7.38e3iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 6.66e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 9.71e4T + 4.75e9T^{2} \) |
| 43 | \( 1 - 1.12e5T + 6.32e9T^{2} \) |
| 47 | \( 1 - 1.96e5T + 1.07e10T^{2} \) |
| 53 | \( 1 - 7.20e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 1.41e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 2.01e5T + 5.15e10T^{2} \) |
| 67 | \( 1 - 2.16e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 3.58e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 4.11e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 2.72e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 2.52e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + 8.72e4T + 4.96e11T^{2} \) |
| 97 | \( 1 + 1.30e5iT - 8.32e11T^{2} \) |
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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
−13.45318451977328838780850406257, −12.36848648281031523917768420943, −11.16833505276637483872576514907, −10.03556272045975489449725306952, −8.706970113175909860167193501831, −8.101533912761670924424029681994, −5.42716147532601393893037433909, −4.24606078712449897836959434784, −2.31052733186608465219307217810, −1.05067235725082536188564168463,
1.88202370985366428029155709075, 4.11479750091565754632974395599, 5.61718730642207482638788399115, 7.12629142290136269428856598785, 8.041327397363663459842192883975, 9.341163015812917720225252999126, 10.47893968801530279537895540898, 12.08842306919051552855086475586, 13.90294718913921152705785420541, 14.19499505462981733528170394681