L(s) = 1 | − 2-s − 1.26·3-s + 4-s − 3.06·5-s + 1.26·6-s + 1.65·7-s − 8-s − 1.39·9-s + 3.06·10-s − 4.09·11-s − 1.26·12-s + 6.45·13-s − 1.65·14-s + 3.88·15-s + 16-s + 1.85·17-s + 1.39·18-s + 8.62·19-s − 3.06·20-s − 2.10·21-s + 4.09·22-s + 5.65·23-s + 1.26·24-s + 4.40·25-s − 6.45·26-s + 5.56·27-s + 1.65·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.731·3-s + 0.5·4-s − 1.37·5-s + 0.517·6-s + 0.626·7-s − 0.353·8-s − 0.464·9-s + 0.970·10-s − 1.23·11-s − 0.365·12-s + 1.79·13-s − 0.442·14-s + 1.00·15-s + 0.250·16-s + 0.449·17-s + 0.328·18-s + 1.97·19-s − 0.685·20-s − 0.458·21-s + 0.872·22-s + 1.17·23-s + 0.258·24-s + 0.881·25-s − 1.26·26-s + 1.07·27-s + 0.313·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7525924759\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7525924759\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 + T \) |
good | 3 | \( 1 + 1.26T + 3T^{2} \) |
| 5 | \( 1 + 3.06T + 5T^{2} \) |
| 7 | \( 1 - 1.65T + 7T^{2} \) |
| 11 | \( 1 + 4.09T + 11T^{2} \) |
| 13 | \( 1 - 6.45T + 13T^{2} \) |
| 17 | \( 1 - 1.85T + 17T^{2} \) |
| 19 | \( 1 - 8.62T + 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 + 3.57T + 29T^{2} \) |
| 37 | \( 1 + 2.30T + 37T^{2} \) |
| 41 | \( 1 + 2.66T + 41T^{2} \) |
| 43 | \( 1 - 0.780T + 43T^{2} \) |
| 47 | \( 1 - 2.91T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 + 4.57T + 59T^{2} \) |
| 61 | \( 1 + 7.87T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 + 7.01T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 + 5.03T + 89T^{2} \) |
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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
−7.989197573473820838940106336214, −7.58547158941671023581220491639, −6.90232144325158598334827689503, −5.79244444391332178722986803142, −5.40669339676730569471294799398, −4.55260598168524906868147370601, −3.40655363405860888829010141843, −2.99971459981524058212862964352, −1.38154374523328291180909089588, −0.58236862954109299565638506390,
0.58236862954109299565638506390, 1.38154374523328291180909089588, 2.99971459981524058212862964352, 3.40655363405860888829010141843, 4.55260598168524906868147370601, 5.40669339676730569471294799398, 5.79244444391332178722986803142, 6.90232144325158598334827689503, 7.58547158941671023581220491639, 7.989197573473820838940106336214