| L(s) = 1 | + i·3-s + (−2.13 + 0.666i)5-s − 9-s + 3.80·11-s + 2.31i·13-s + (−0.666 − 2.13i)15-s − 0.375i·17-s − 0.906·19-s − 1.31i·23-s + (4.11 − 2.84i)25-s − i·27-s + 6.15·29-s + 5.62·31-s + 3.80i·33-s − 5.73i·37-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.954 + 0.297i)5-s − 0.333·9-s + 1.14·11-s + 0.641i·13-s + (−0.171 − 0.551i)15-s − 0.0909i·17-s − 0.207·19-s − 0.273i·23-s + (0.822 − 0.568i)25-s − 0.192i·27-s + 1.14·29-s + 1.01·31-s + 0.661i·33-s − 0.943i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.297 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.297 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.353106547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.353106547\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (2.13 - 0.666i)T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 3.80T + 11T^{2} \) |
| 13 | \( 1 - 2.31iT - 13T^{2} \) |
| 17 | \( 1 + 0.375iT - 17T^{2} \) |
| 19 | \( 1 + 0.906T + 19T^{2} \) |
| 23 | \( 1 + 1.31iT - 23T^{2} \) |
| 29 | \( 1 - 6.15T + 29T^{2} \) |
| 31 | \( 1 - 5.62T + 31T^{2} \) |
| 37 | \( 1 + 5.73iT - 37T^{2} \) |
| 41 | \( 1 - 0.199T + 41T^{2} \) |
| 43 | \( 1 - 12.4iT - 43T^{2} \) |
| 47 | \( 1 - 10.8iT - 47T^{2} \) |
| 53 | \( 1 - 3.06iT - 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 - 2.17T + 61T^{2} \) |
| 67 | \( 1 + 5.56iT - 67T^{2} \) |
| 71 | \( 1 + 9.88T + 71T^{2} \) |
| 73 | \( 1 - 6.40iT - 73T^{2} \) |
| 79 | \( 1 - 15.2T + 79T^{2} \) |
| 83 | \( 1 + 8.99iT - 83T^{2} \) |
| 89 | \( 1 - 5.01T + 89T^{2} \) |
| 97 | \( 1 - 6.36iT - 97T^{2} \) |
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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
−9.083539901243099809944249569919, −8.222740976716848546898927344509, −7.57628789744414434857191848509, −6.54569501378429749533835503146, −6.20124795304627624571152035465, −4.68736263804635460425960660776, −4.38974421343481585736672196519, −3.49056970473509362709552445738, −2.63474528376224121120942207876, −1.10880521615876091993608503042,
0.51903950165597492538725600859, 1.54575896623083465534141640779, 2.89747245182840935127197851846, 3.73173283886278570156912004279, 4.55669309343017476940456198771, 5.45062554145120632908130376886, 6.48106600982678283438204787281, 6.96896742165581598404985289825, 7.86612653458887880567751741568, 8.443360646207495294908079368340
