L(s) = 1 | + 0.224·3-s − 2.01·5-s − 1.98·7-s − 2.94·9-s − 5.16·11-s + 4.72·13-s − 0.451·15-s + 3.77·17-s − 1.20·19-s − 0.445·21-s + 1.63·23-s − 0.951·25-s − 1.33·27-s − 8.63·29-s − 6.56·31-s − 1.16·33-s + 3.98·35-s + 1.73·37-s + 1.06·39-s + 0.973·41-s + 6.75·43-s + 5.93·45-s − 11.9·47-s − 3.06·49-s + 0.848·51-s − 8.24·53-s + 10.3·55-s + ⋯ |
L(s) = 1 | + 0.129·3-s − 0.899·5-s − 0.749·7-s − 0.983·9-s − 1.55·11-s + 1.30·13-s − 0.116·15-s + 0.915·17-s − 0.276·19-s − 0.0971·21-s + 0.340·23-s − 0.190·25-s − 0.257·27-s − 1.60·29-s − 1.17·31-s − 0.201·33-s + 0.674·35-s + 0.284·37-s + 0.169·39-s + 0.151·41-s + 1.02·43-s + 0.884·45-s − 1.74·47-s − 0.438·49-s + 0.118·51-s − 1.13·53-s + 1.40·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5527569011\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5527569011\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
good | 3 | \( 1 - 0.224T + 3T^{2} \) |
| 5 | \( 1 + 2.01T + 5T^{2} \) |
| 7 | \( 1 + 1.98T + 7T^{2} \) |
| 11 | \( 1 + 5.16T + 11T^{2} \) |
| 13 | \( 1 - 4.72T + 13T^{2} \) |
| 17 | \( 1 - 3.77T + 17T^{2} \) |
| 19 | \( 1 + 1.20T + 19T^{2} \) |
| 23 | \( 1 - 1.63T + 23T^{2} \) |
| 29 | \( 1 + 8.63T + 29T^{2} \) |
| 31 | \( 1 + 6.56T + 31T^{2} \) |
| 37 | \( 1 - 1.73T + 37T^{2} \) |
| 41 | \( 1 - 0.973T + 41T^{2} \) |
| 43 | \( 1 - 6.75T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 + 8.24T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 - 1.56T + 61T^{2} \) |
| 67 | \( 1 - 8.49T + 67T^{2} \) |
| 71 | \( 1 - 0.0603T + 71T^{2} \) |
| 73 | \( 1 + 3.47T + 73T^{2} \) |
| 79 | \( 1 - 6.07T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 - 0.498T + 89T^{2} \) |
| 97 | \( 1 + 8.81T + 97T^{2} \) |
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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
−7.900268392419262929704744259025, −7.37292731378093520428039924364, −6.33893285693372080954905354360, −5.73390008680140050117931135421, −5.19766907297794744117944294309, −4.08858255552345737098985169280, −3.33264891913536988192524566496, −3.02005031918973557736866573305, −1.81840274505535517351576605699, −0.34731497961269972225936524578,
0.34731497961269972225936524578, 1.81840274505535517351576605699, 3.02005031918973557736866573305, 3.33264891913536988192524566496, 4.08858255552345737098985169280, 5.19766907297794744117944294309, 5.73390008680140050117931135421, 6.33893285693372080954905354360, 7.37292731378093520428039924364, 7.900268392419262929704744259025