L(s) = 1 | + 2-s − 3-s + 4-s − 4.06·5-s − 6-s + 4.31·7-s + 8-s + 9-s − 4.06·10-s − 3.45·11-s − 12-s − 13-s + 4.31·14-s + 4.06·15-s + 16-s − 7.56·17-s + 18-s − 1.84·19-s − 4.06·20-s − 4.31·21-s − 3.45·22-s − 8.50·23-s − 24-s + 11.5·25-s − 26-s − 27-s + 4.31·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.81·5-s − 0.408·6-s + 1.63·7-s + 0.353·8-s + 0.333·9-s − 1.28·10-s − 1.04·11-s − 0.288·12-s − 0.277·13-s + 1.15·14-s + 1.05·15-s + 0.250·16-s − 1.83·17-s + 0.235·18-s − 0.424·19-s − 0.909·20-s − 0.942·21-s − 0.737·22-s − 1.77·23-s − 0.204·24-s + 2.31·25-s − 0.196·26-s − 0.192·27-s + 0.816·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.317558521\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.317558521\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
good | 5 | \( 1 + 4.06T + 5T^{2} \) |
| 7 | \( 1 - 4.31T + 7T^{2} \) |
| 11 | \( 1 + 3.45T + 11T^{2} \) |
| 17 | \( 1 + 7.56T + 17T^{2} \) |
| 19 | \( 1 + 1.84T + 19T^{2} \) |
| 23 | \( 1 + 8.50T + 23T^{2} \) |
| 29 | \( 1 - 6.61T + 29T^{2} \) |
| 31 | \( 1 + 5.55T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 - 0.695T + 41T^{2} \) |
| 43 | \( 1 + 3.09T + 43T^{2} \) |
| 47 | \( 1 + 1.17T + 47T^{2} \) |
| 53 | \( 1 - 5.76T + 53T^{2} \) |
| 59 | \( 1 - 6.88T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 + 14.3T + 67T^{2} \) |
| 71 | \( 1 - 4.40T + 71T^{2} \) |
| 73 | \( 1 + 8.29T + 73T^{2} \) |
| 79 | \( 1 - 6.21T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 + 2.88T + 89T^{2} \) |
| 97 | \( 1 - 4.85T + 97T^{2} \) |
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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
−7.76872753740719835976623666381, −7.23828203036481850885397743566, −6.46751677833063377020521940435, −5.52445448314367404135248009753, −4.78556584833424349318372158404, −4.35336632450483268336468538770, −3.98049302389213811035828656360, −2.67727364094744646549525183990, −1.92852141511718703416718700533, −0.50635352261230398466069064662,
0.50635352261230398466069064662, 1.92852141511718703416718700533, 2.67727364094744646549525183990, 3.98049302389213811035828656360, 4.35336632450483268336468538770, 4.78556584833424349318372158404, 5.52445448314367404135248009753, 6.46751677833063377020521940435, 7.23828203036481850885397743566, 7.76872753740719835976623666381