| L(s) = 1 | + 5-s + 4·11-s + 2·13-s + 2·17-s + 4·19-s + 25-s + 10·29-s + 6·37-s − 6·41-s − 4·43-s − 8·47-s − 6·53-s + 4·55-s − 4·59-s + 10·61-s + 2·65-s + 4·67-s + 16·71-s + 14·73-s + 8·79-s − 4·83-s + 2·85-s + 10·89-s + 4·95-s − 10·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.20·11-s + 0.554·13-s + 0.485·17-s + 0.917·19-s + 1/5·25-s + 1.85·29-s + 0.986·37-s − 0.937·41-s − 0.609·43-s − 1.16·47-s − 0.824·53-s + 0.539·55-s − 0.520·59-s + 1.28·61-s + 0.248·65-s + 0.488·67-s + 1.89·71-s + 1.63·73-s + 0.900·79-s − 0.439·83-s + 0.216·85-s + 1.05·89-s + 0.410·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.294934498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.294934498\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−15.84846309396613, −15.30089811222655, −14.62369198236104, −14.07448625827142, −13.82467297567137, −13.13625868238631, −12.42224866386466, −11.99293421952462, −11.37332774452769, −10.91776965020614, −9.991700813182238, −9.727538008886639, −9.153693127693244, −8.314209685802651, −8.066295320465384, −7.015533039732564, −6.526450237963286, −6.113159076847965, −5.189528224150418, −4.747927544785832, −3.750045335708403, −3.313235396192037, −2.393530791661948, −1.424984871400902, −0.8626797252492385,
0.8626797252492385, 1.424984871400902, 2.393530791661948, 3.313235396192037, 3.750045335708403, 4.747927544785832, 5.189528224150418, 6.113159076847965, 6.526450237963286, 7.015533039732564, 8.066295320465384, 8.314209685802651, 9.153693127693244, 9.727538008886639, 9.991700813182238, 10.91776965020614, 11.37332774452769, 11.99293421952462, 12.42224866386466, 13.13625868238631, 13.82467297567137, 14.07448625827142, 14.62369198236104, 15.30089811222655, 15.84846309396613
