L(s) = 1 | + 2-s + 1.17·3-s + 4-s + 5-s + 1.17·6-s + 7-s + 8-s − 1.62·9-s + 10-s + 1.17·12-s + 2.71·13-s + 14-s + 1.17·15-s + 16-s − 5.34·17-s − 1.62·18-s + 3.97·19-s + 20-s + 1.17·21-s + 5.49·23-s + 1.17·24-s + 25-s + 2.71·26-s − 5.42·27-s + 28-s − 4.99·29-s + 1.17·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.676·3-s + 0.5·4-s + 0.447·5-s + 0.478·6-s + 0.377·7-s + 0.353·8-s − 0.542·9-s + 0.316·10-s + 0.338·12-s + 0.752·13-s + 0.267·14-s + 0.302·15-s + 0.250·16-s − 1.29·17-s − 0.383·18-s + 0.912·19-s + 0.223·20-s + 0.255·21-s + 1.14·23-s + 0.239·24-s + 0.200·25-s + 0.532·26-s − 1.04·27-s + 0.188·28-s − 0.928·29-s + 0.213·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.994554368\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.994554368\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 - 1.17T + 3T^{2} \) |
| 13 | \( 1 - 2.71T + 13T^{2} \) |
| 17 | \( 1 + 5.34T + 17T^{2} \) |
| 19 | \( 1 - 3.97T + 19T^{2} \) |
| 23 | \( 1 - 5.49T + 23T^{2} \) |
| 29 | \( 1 + 4.99T + 29T^{2} \) |
| 31 | \( 1 + 2.87T + 31T^{2} \) |
| 37 | \( 1 - 0.0405T + 37T^{2} \) |
| 41 | \( 1 - 5.13T + 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 - 9.89T + 47T^{2} \) |
| 53 | \( 1 - 4.55T + 53T^{2} \) |
| 59 | \( 1 + 0.584T + 59T^{2} \) |
| 61 | \( 1 - 7.25T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 - 8.20T + 71T^{2} \) |
| 73 | \( 1 - 14.5T + 73T^{2} \) |
| 79 | \( 1 - 6.75T + 79T^{2} \) |
| 83 | \( 1 + 15.5T + 83T^{2} \) |
| 89 | \( 1 - 2.73T + 89T^{2} \) |
| 97 | \( 1 + 1.32T + 97T^{2} \) |
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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
−7.58290171430952807437469058636, −7.22273563827123455657286160434, −6.22398142234693188231185720162, −5.69183434433736288990285009581, −5.05222417946228145654731974017, −4.14074463287674760005796825849, −3.52138588675420794831739621030, −2.62494446030834530012334892286, −2.11773170978906420519775541595, −0.971382045494201983280980560569,
0.971382045494201983280980560569, 2.11773170978906420519775541595, 2.62494446030834530012334892286, 3.52138588675420794831739621030, 4.14074463287674760005796825849, 5.05222417946228145654731974017, 5.69183434433736288990285009581, 6.22398142234693188231185720162, 7.22273563827123455657286160434, 7.58290171430952807437469058636