L(s) = 1 | + 1.78·2-s − 3-s + 1.17·4-s − 2.21·5-s − 1.78·6-s − 7-s − 1.47·8-s + 9-s − 3.94·10-s − 0.953·11-s − 1.17·12-s − 2.99·13-s − 1.78·14-s + 2.21·15-s − 4.97·16-s − 1.19·17-s + 1.78·18-s + 2.34·19-s − 2.59·20-s + 21-s − 1.69·22-s + 7.39·23-s + 1.47·24-s − 0.0893·25-s − 5.32·26-s − 27-s − 1.17·28-s + ⋯ |
L(s) = 1 | + 1.25·2-s − 0.577·3-s + 0.585·4-s − 0.991·5-s − 0.726·6-s − 0.377·7-s − 0.522·8-s + 0.333·9-s − 1.24·10-s − 0.287·11-s − 0.337·12-s − 0.829·13-s − 0.475·14-s + 0.572·15-s − 1.24·16-s − 0.290·17-s + 0.419·18-s + 0.538·19-s − 0.579·20-s + 0.218·21-s − 0.361·22-s + 1.54·23-s + 0.301·24-s − 0.0178·25-s − 1.04·26-s − 0.192·27-s − 0.221·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.464151555\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.464151555\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 - T \) |
good | 2 | \( 1 - 1.78T + 2T^{2} \) |
| 5 | \( 1 + 2.21T + 5T^{2} \) |
| 11 | \( 1 + 0.953T + 11T^{2} \) |
| 13 | \( 1 + 2.99T + 13T^{2} \) |
| 17 | \( 1 + 1.19T + 17T^{2} \) |
| 19 | \( 1 - 2.34T + 19T^{2} \) |
| 23 | \( 1 - 7.39T + 23T^{2} \) |
| 29 | \( 1 + 10.5T + 29T^{2} \) |
| 31 | \( 1 - 5.54T + 31T^{2} \) |
| 37 | \( 1 - 4.22T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 2.05T + 47T^{2} \) |
| 53 | \( 1 - 8.12T + 53T^{2} \) |
| 59 | \( 1 - 9.48T + 59T^{2} \) |
| 61 | \( 1 + 4.79T + 61T^{2} \) |
| 67 | \( 1 - 6.29T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 6.76T + 73T^{2} \) |
| 79 | \( 1 + 1.95T + 79T^{2} \) |
| 83 | \( 1 + 5.95T + 83T^{2} \) |
| 89 | \( 1 + 4.49T + 89T^{2} \) |
| 97 | \( 1 - 1.11T + 97T^{2} \) |
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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
−8.335422565995409497910473063892, −7.28082011651732863604766597892, −7.03514885702196119223765507008, −5.96704502212251077283606327986, −5.33165169113024435616066920417, −4.69164171555991529825007951622, −3.96941758526438811696909653375, −3.27240220316107884856321594404, −2.36703201900020277887966563590, −0.56375957725151085355618881822,
0.56375957725151085355618881822, 2.36703201900020277887966563590, 3.27240220316107884856321594404, 3.96941758526438811696909653375, 4.69164171555991529825007951622, 5.33165169113024435616066920417, 5.96704502212251077283606327986, 7.03514885702196119223765507008, 7.28082011651732863604766597892, 8.335422565995409497910473063892