| L(s) = 1 | + 3i·3-s − 2i·7-s − 6·9-s − i·13-s + 6·21-s − i·23-s − 9i·27-s + 3·29-s + 3·31-s − 8i·37-s + 3·39-s + 3·41-s + 2i·43-s − 11i·47-s + 3·49-s + ⋯ |
| L(s) = 1 | + 1.73i·3-s − 0.755i·7-s − 2·9-s − 0.277i·13-s + 1.30·21-s − 0.208i·23-s − 1.73i·27-s + 0.557·29-s + 0.538·31-s − 1.31i·37-s + 0.480·39-s + 0.468·41-s + 0.304i·43-s − 1.60i·47-s + 0.428·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.731287754\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.731287754\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + iT \) |
| good | 3 | \( 1 - 3iT - 3T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 + 11iT - 47T^{2} \) |
| 53 | \( 1 - 14iT - 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 7T + 71T^{2} \) |
| 73 | \( 1 - 9iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 18iT - 97T^{2} \) |
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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
−8.662486920376952591533446037602, −7.85205156031396279168738253681, −7.02404820238043737778499722383, −6.04178741119857415136783117697, −5.34321068353149304637587292023, −4.57194838248218024187446697513, −4.01931324585700801304247440971, −3.34227735495195328919822928698, −2.40255080554156027422688666389, −0.69843220659036803517831161922,
0.76528491901308749605670720691, 1.76183632493648659148647100219, 2.49210468537196315234766297334, 3.27879554403493564280515281657, 4.61060177125585013665299574561, 5.51064474057715498178683223258, 6.22224997531827549575503713729, 6.70273101460077679967451138801, 7.45314201601922274236887331474, 8.149097666441831610179014039050
