L(s) = 1 | + 2-s − 3-s + 4-s + 2.93·5-s − 6-s + 7-s + 8-s + 9-s + 2.93·10-s + 2.81·11-s − 12-s + 14-s − 2.93·15-s + 16-s + 4.00·17-s + 18-s + 6.01·19-s + 2.93·20-s − 21-s + 2.81·22-s − 2.16·23-s − 24-s + 3.60·25-s − 27-s + 28-s + 7.58·29-s − 2.93·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.31·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.927·10-s + 0.847·11-s − 0.288·12-s + 0.267·14-s − 0.757·15-s + 0.250·16-s + 0.972·17-s + 0.235·18-s + 1.38·19-s + 0.656·20-s − 0.218·21-s + 0.599·22-s − 0.451·23-s − 0.204·24-s + 0.721·25-s − 0.192·27-s + 0.188·28-s + 1.40·29-s − 0.535·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.465678331\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.465678331\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 2.93T + 5T^{2} \) |
| 11 | \( 1 - 2.81T + 11T^{2} \) |
| 17 | \( 1 - 4.00T + 17T^{2} \) |
| 19 | \( 1 - 6.01T + 19T^{2} \) |
| 23 | \( 1 + 2.16T + 23T^{2} \) |
| 29 | \( 1 - 7.58T + 29T^{2} \) |
| 31 | \( 1 - 3.12T + 31T^{2} \) |
| 37 | \( 1 - 1.86T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 + 3.37T + 43T^{2} \) |
| 47 | \( 1 + 3.99T + 47T^{2} \) |
| 53 | \( 1 + 2.55T + 53T^{2} \) |
| 59 | \( 1 + 1.79T + 59T^{2} \) |
| 61 | \( 1 - 4.47T + 61T^{2} \) |
| 67 | \( 1 + 2.80T + 67T^{2} \) |
| 71 | \( 1 - 3.55T + 71T^{2} \) |
| 73 | \( 1 + 9.93T + 73T^{2} \) |
| 79 | \( 1 + 8.96T + 79T^{2} \) |
| 83 | \( 1 - 3.43T + 83T^{2} \) |
| 89 | \( 1 + 8.84T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−7.78912727254072064449908035592, −6.93250062738389897289682229422, −6.36096907824694008202547094360, −5.78064875454034599044335280192, −5.17706398554108328768196925404, −4.61385769601739343772153222669, −3.57230438368896309480970566293, −2.79452831936626209479529887157, −1.68051928600538306692929783793, −1.13162578513058941997768594804,
1.13162578513058941997768594804, 1.68051928600538306692929783793, 2.79452831936626209479529887157, 3.57230438368896309480970566293, 4.61385769601739343772153222669, 5.17706398554108328768196925404, 5.78064875454034599044335280192, 6.36096907824694008202547094360, 6.93250062738389897289682229422, 7.78912727254072064449908035592