L(s) = 1 | + 2·3-s + 5-s + 9-s + 4·11-s − 2·13-s + 2·15-s + 7·17-s + 19-s − 3·23-s − 4·25-s − 4·27-s + 4·29-s − 4·31-s + 8·33-s + 10·37-s − 4·39-s − 9·43-s + 45-s + 4·47-s + 14·51-s + 6·53-s + 4·55-s + 2·57-s + 12·59-s + 2·61-s − 2·65-s + 16·67-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.516·15-s + 1.69·17-s + 0.229·19-s − 0.625·23-s − 4/5·25-s − 0.769·27-s + 0.742·29-s − 0.718·31-s + 1.39·33-s + 1.64·37-s − 0.640·39-s − 1.37·43-s + 0.149·45-s + 0.583·47-s + 1.96·51-s + 0.824·53-s + 0.539·55-s + 0.264·57-s + 1.56·59-s + 0.256·61-s − 0.248·65-s + 1.95·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.838669700\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.838669700\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−8.025401627931226792422643584924, −7.35576368051028550749728113220, −6.56908747562290589936217181967, −5.78918965685504123439080716994, −5.15668857123074570419802384411, −3.95758752330729585297373817721, −3.61004359162261295691312720452, −2.65476733358185634518476010275, −1.96150973465548707976363407788, −0.968935721867244998007328451044,
0.968935721867244998007328451044, 1.96150973465548707976363407788, 2.65476733358185634518476010275, 3.61004359162261295691312720452, 3.95758752330729585297373817721, 5.15668857123074570419802384411, 5.78918965685504123439080716994, 6.56908747562290589936217181967, 7.35576368051028550749728113220, 8.025401627931226792422643584924