L(s) = 1 | + 0.618·2-s − 3-s − 1.61·4-s − 5-s − 0.618·6-s − 3.23·7-s − 2.23·8-s + 9-s − 0.618·10-s + 1.61·12-s − 5.23·13-s − 2.00·14-s + 15-s + 1.85·16-s + 5.47·17-s + 0.618·18-s − 6.47·19-s + 1.61·20-s + 3.23·21-s − 4.70·23-s + 2.23·24-s + 25-s − 3.23·26-s − 27-s + 5.23·28-s + 1.23·29-s + 0.618·30-s + ⋯ |
L(s) = 1 | + 0.437·2-s − 0.577·3-s − 0.809·4-s − 0.447·5-s − 0.252·6-s − 1.22·7-s − 0.790·8-s + 0.333·9-s − 0.195·10-s + 0.467·12-s − 1.45·13-s − 0.534·14-s + 0.258·15-s + 0.463·16-s + 1.32·17-s + 0.145·18-s − 1.48·19-s + 0.361·20-s + 0.706·21-s − 0.981·23-s + 0.456·24-s + 0.200·25-s − 0.634·26-s − 0.192·27-s + 0.989·28-s + 0.229·29-s + 0.112·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5249274860\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5249274860\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 7 | \( 1 + 3.23T + 7T^{2} \) |
| 13 | \( 1 + 5.23T + 13T^{2} \) |
| 17 | \( 1 - 5.47T + 17T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 23 | \( 1 + 4.70T + 23T^{2} \) |
| 29 | \( 1 - 1.23T + 29T^{2} \) |
| 31 | \( 1 + 6.70T + 31T^{2} \) |
| 37 | \( 1 - 0.763T + 37T^{2} \) |
| 41 | \( 1 + 3.52T + 41T^{2} \) |
| 43 | \( 1 - 5.23T + 43T^{2} \) |
| 47 | \( 1 - 8.70T + 47T^{2} \) |
| 53 | \( 1 - 9.94T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 - 1.47T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 4.76T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−9.427735204239699058008387852557, −8.527456742297054131306434749811, −7.58416251207348191836523596488, −6.78981741597787002403718131194, −5.86116861770557489530215626444, −5.27108553574512450169205395726, −4.21676799355029820028982200029, −3.65502474291857546106345561192, −2.49151405356455115431622926798, −0.45256107468636918978321523588,
0.45256107468636918978321523588, 2.49151405356455115431622926798, 3.65502474291857546106345561192, 4.21676799355029820028982200029, 5.27108553574512450169205395726, 5.86116861770557489530215626444, 6.78981741597787002403718131194, 7.58416251207348191836523596488, 8.527456742297054131306434749811, 9.427735204239699058008387852557