| L(s) = 1 | + 3i·3-s + 32i·7-s − 9·9-s + 36·11-s + 10i·13-s − 78i·17-s − 140·19-s − 96·21-s + 192i·23-s − 27i·27-s − 6·29-s − 16·31-s + 108i·33-s − 34i·37-s − 30·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 1.72i·7-s − 0.333·9-s + 0.986·11-s + 0.213i·13-s − 1.11i·17-s − 1.69·19-s − 0.997·21-s + 1.74i·23-s − 0.192i·27-s − 0.0384·29-s − 0.0926·31-s + 0.569i·33-s − 0.151i·37-s − 0.123·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.249069296\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.249069296\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 32iT - 343T^{2} \) |
| 11 | \( 1 - 36T + 1.33e3T^{2} \) |
| 13 | \( 1 - 10iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 78iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 140T + 6.85e3T^{2} \) |
| 23 | \( 1 - 192iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 16T + 2.97e4T^{2} \) |
| 37 | \( 1 + 34iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 390T + 6.89e4T^{2} \) |
| 43 | \( 1 - 52iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 408iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 114iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 516T + 2.05e5T^{2} \) |
| 61 | \( 1 + 58T + 2.26e5T^{2} \) |
| 67 | \( 1 + 892iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 120T + 3.57e5T^{2} \) |
| 73 | \( 1 - 646iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 732iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.59e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 194iT - 9.12e5T^{2} \) |
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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
−11.75307123070010315173154494862, −10.90725182594560341120921138472, −9.411476244455909597423906338065, −9.204288583641642825599473112711, −8.125320288630902359250080874159, −6.61932832013985628643186946535, −5.69545931301681462303955904762, −4.66716437551174447110892438396, −3.28836453152729999704062405703, −1.97244379491445672001452966247,
0.45303732462648160499282063648, 1.79576518050718373837789698804, 3.67082297408562190581801166016, 4.52385835277050934959864041572, 6.36392499259782491146598848347, 6.81047141401893679036521667879, 8.005441945174035097073673731921, 8.800520609734089177588183491482, 10.36467604784341250172957116832, 10.64920102579524409541995821930
