L(s) = 1 | + 2-s + 2·3-s + 4-s − 5-s + 2·6-s + 8-s + 9-s − 10-s + 2·12-s − 13-s − 2·15-s + 16-s + 18-s − 20-s + 2·23-s + 2·24-s + 25-s − 26-s − 4·27-s − 8·29-s − 2·30-s + 4·31-s + 32-s + 36-s − 2·37-s − 2·39-s − 40-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.447·5-s + 0.816·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.577·12-s − 0.277·13-s − 0.516·15-s + 1/4·16-s + 0.235·18-s − 0.223·20-s + 0.417·23-s + 0.408·24-s + 1/5·25-s − 0.196·26-s − 0.769·27-s − 1.48·29-s − 0.365·30-s + 0.718·31-s + 0.176·32-s + 1/6·36-s − 0.328·37-s − 0.320·39-s − 0.158·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.995324859\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.995324859\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.74270768272437, −14.43198125748492, −13.90120916324764, −13.27689819037968, −12.99372716490511, −12.34725949253058, −11.81257767667681, −11.21443210820244, −10.80984941587338, −10.04712149916799, −9.382680749301880, −9.052771196712770, −8.303470258756353, −7.822480521996436, −7.418124161733218, −6.761907016132581, −6.090889485603534, −5.379461793939603, −4.825053236337642, −3.996475526584893, −3.671694787975600, −2.968281510626410, −2.406036636208891, −1.774486842912306, −0.6606750439330661,
0.6606750439330661, 1.774486842912306, 2.406036636208891, 2.968281510626410, 3.671694787975600, 3.996475526584893, 4.825053236337642, 5.379461793939603, 6.090889485603534, 6.761907016132581, 7.418124161733218, 7.822480521996436, 8.303470258756353, 9.052771196712770, 9.382680749301880, 10.04712149916799, 10.80984941587338, 11.21443210820244, 11.81257767667681, 12.34725949253058, 12.99372716490511, 13.27689819037968, 13.90120916324764, 14.43198125748492, 14.74270768272437