| L(s) = 1 | − i·3-s + (−1.52 + 1.63i)5-s + 2.82i·7-s − 9-s − 1.05·11-s − i·13-s + (1.63 + 1.52i)15-s + 7.14i·17-s − 6.61·19-s + 2.82·21-s − 3.49i·23-s + (−0.332 − 4.98i)25-s + i·27-s − 4.82·29-s + 9.65·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (−0.683 + 0.730i)5-s + 1.06i·7-s − 0.333·9-s − 0.318·11-s − 0.277i·13-s + (0.421 + 0.394i)15-s + 1.73i·17-s − 1.51·19-s + 0.617·21-s − 0.728i·23-s + (−0.0664 − 0.997i)25-s + 0.192i·27-s − 0.896·29-s + 1.73·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.683 + 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.683 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1948851270\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1948851270\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.52 - 1.63i)T \) |
| 13 | \( 1 + iT \) |
| good | 7 | \( 1 - 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 1.05T + 11T^{2} \) |
| 17 | \( 1 - 7.14iT - 17T^{2} \) |
| 19 | \( 1 + 6.61T + 19T^{2} \) |
| 23 | \( 1 + 3.49iT - 23T^{2} \) |
| 29 | \( 1 + 4.82T + 29T^{2} \) |
| 31 | \( 1 - 9.65T + 31T^{2} \) |
| 37 | \( 1 - 8.11iT - 37T^{2} \) |
| 41 | \( 1 + 3.26T + 41T^{2} \) |
| 43 | \( 1 + 7.44iT - 43T^{2} \) |
| 47 | \( 1 + 11.3iT - 47T^{2} \) |
| 53 | \( 1 + 4.53iT - 53T^{2} \) |
| 59 | \( 1 - 1.05T + 59T^{2} \) |
| 61 | \( 1 + 7.97T + 61T^{2} \) |
| 67 | \( 1 + 1.46iT - 67T^{2} \) |
| 71 | \( 1 + 0.845T + 71T^{2} \) |
| 73 | \( 1 + 9.28iT - 73T^{2} \) |
| 79 | \( 1 - 6.85T + 79T^{2} \) |
| 83 | \( 1 - 7.97iT - 83T^{2} \) |
| 89 | \( 1 + 0.139T + 89T^{2} \) |
| 97 | \( 1 + 6.93iT - 97T^{2} \) |
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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
−8.402407361315103793124230735030, −7.85311718455695404659388256840, −6.69035395019837956672926682602, −6.37501365657182000387913973884, −5.54088866960617883482889753005, −4.45172146515037995679086453012, −3.54343448025242013807414623992, −2.58896335430893605521109856622, −1.87809108011991915347181844556, −0.06676320228470484738520541032,
1.06623437779932593396757941156, 2.59083435534561477255637643749, 3.64672352181339885601468195452, 4.42632356399380836381523050340, 4.78786738971404626286941693017, 5.83083046546918505706731291634, 6.86215367370877749295511865189, 7.59874935111427237945299584349, 8.131590339311971337344660529255, 9.134697437031830259140441140198
