L(s) = 1 | − 3-s + 0.754·5-s − 4.03·7-s + 9-s − 3.82·11-s − 4.25·13-s − 0.754·15-s − 5.34·17-s − 3.06·19-s + 4.03·21-s − 3.68·23-s − 4.43·25-s − 27-s − 6.36·29-s − 10.0·31-s + 3.82·33-s − 3.04·35-s + 9.28·37-s + 4.25·39-s + 2.62·41-s + 7.26·43-s + 0.754·45-s − 10.3·47-s + 9.26·49-s + 5.34·51-s + 10.2·53-s − 2.88·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.337·5-s − 1.52·7-s + 0.333·9-s − 1.15·11-s − 1.18·13-s − 0.194·15-s − 1.29·17-s − 0.704·19-s + 0.880·21-s − 0.768·23-s − 0.886·25-s − 0.192·27-s − 1.18·29-s − 1.81·31-s + 0.665·33-s − 0.514·35-s + 1.52·37-s + 0.681·39-s + 0.410·41-s + 1.10·43-s + 0.112·45-s − 1.51·47-s + 1.32·49-s + 0.748·51-s + 1.41·53-s − 0.388·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1384090128\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1384090128\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 503 | \( 1 + T \) |
good | 5 | \( 1 - 0.754T + 5T^{2} \) |
| 7 | \( 1 + 4.03T + 7T^{2} \) |
| 11 | \( 1 + 3.82T + 11T^{2} \) |
| 13 | \( 1 + 4.25T + 13T^{2} \) |
| 17 | \( 1 + 5.34T + 17T^{2} \) |
| 19 | \( 1 + 3.06T + 19T^{2} \) |
| 23 | \( 1 + 3.68T + 23T^{2} \) |
| 29 | \( 1 + 6.36T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 - 9.28T + 37T^{2} \) |
| 41 | \( 1 - 2.62T + 41T^{2} \) |
| 43 | \( 1 - 7.26T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 - 6.42T + 59T^{2} \) |
| 61 | \( 1 - 9.19T + 61T^{2} \) |
| 67 | \( 1 + 6.71T + 67T^{2} \) |
| 71 | \( 1 + 0.934T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 + 0.170T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 - 7.13T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−7.87746826898543200205080734441, −7.26765656075648461113320381513, −6.62607887149198109910807715534, −5.85508906893820154363437362763, −5.46742731492026397377685324101, −4.41579524047146805170931929120, −3.74753260074943449756463508751, −2.55729800723850033258404445788, −2.11838261420643982407086584116, −0.18253035624982413178492872370,
0.18253035624982413178492872370, 2.11838261420643982407086584116, 2.55729800723850033258404445788, 3.74753260074943449756463508751, 4.41579524047146805170931929120, 5.46742731492026397377685324101, 5.85508906893820154363437362763, 6.62607887149198109910807715534, 7.26765656075648461113320381513, 7.87746826898543200205080734441