L(s) = 1 | − 2-s + 3.41·3-s + 4-s + 5-s − 3.41·6-s − 8-s + 8.65·9-s − 10-s − 0.828·11-s + 3.41·12-s − 4.82·13-s + 3.41·15-s + 16-s + 2.58·17-s − 8.65·18-s + 0.585·19-s + 20-s + 0.828·22-s − 1.17·23-s − 3.41·24-s + 25-s + 4.82·26-s + 19.3·27-s − 4.82·29-s − 3.41·30-s − 2.82·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.97·3-s + 0.5·4-s + 0.447·5-s − 1.39·6-s − 0.353·8-s + 2.88·9-s − 0.316·10-s − 0.249·11-s + 0.985·12-s − 1.33·13-s + 0.881·15-s + 0.250·16-s + 0.627·17-s − 2.04·18-s + 0.134·19-s + 0.223·20-s + 0.176·22-s − 0.244·23-s − 0.696·24-s + 0.200·25-s + 0.946·26-s + 3.71·27-s − 0.896·29-s − 0.623·30-s − 0.508·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.071613186\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.071613186\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 3.41T + 3T^{2} \) |
| 11 | \( 1 + 0.828T + 11T^{2} \) |
| 13 | \( 1 + 4.82T + 13T^{2} \) |
| 17 | \( 1 - 2.58T + 17T^{2} \) |
| 19 | \( 1 - 0.585T + 19T^{2} \) |
| 23 | \( 1 + 1.17T + 23T^{2} \) |
| 29 | \( 1 + 4.82T + 29T^{2} \) |
| 31 | \( 1 + 2.82T + 31T^{2} \) |
| 37 | \( 1 + 7.65T + 37T^{2} \) |
| 41 | \( 1 + 3.07T + 41T^{2} \) |
| 43 | \( 1 + 8.82T + 43T^{2} \) |
| 47 | \( 1 - 5.17T + 47T^{2} \) |
| 53 | \( 1 - 6.48T + 53T^{2} \) |
| 59 | \( 1 - 8.58T + 59T^{2} \) |
| 61 | \( 1 - 9.31T + 61T^{2} \) |
| 67 | \( 1 - 1.65T + 67T^{2} \) |
| 71 | \( 1 + 4.48T + 71T^{2} \) |
| 73 | \( 1 + 9.41T + 73T^{2} \) |
| 79 | \( 1 + 6.82T + 79T^{2} \) |
| 83 | \( 1 + 2.24T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + 7.75T + 97T^{2} \) |
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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
−10.28971780065301490311421084737, −9.943249755000821756197060500778, −9.117572308403497337302090990962, −8.410575894350371954381270306616, −7.51000711774831488048519746298, −6.97136286400532135914052356201, −5.21403970872239700718804396046, −3.73518003555372199495859711444, −2.66773972862876738845706573219, −1.77576856829648359670824497850,
1.77576856829648359670824497850, 2.66773972862876738845706573219, 3.73518003555372199495859711444, 5.21403970872239700718804396046, 6.97136286400532135914052356201, 7.51000711774831488048519746298, 8.410575894350371954381270306616, 9.117572308403497337302090990962, 9.943249755000821756197060500778, 10.28971780065301490311421084737