| L(s) = 1 | + 5-s − 13-s + 2·17-s + 25-s + 2·29-s + 4·31-s + 6·37-s − 6·41-s + 4·43-s − 4·47-s + 10·53-s + 2·61-s − 65-s + 8·67-s − 4·71-s + 6·73-s − 8·79-s + 8·83-s + 2·85-s − 6·89-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.277·13-s + 0.485·17-s + 1/5·25-s + 0.371·29-s + 0.718·31-s + 0.986·37-s − 0.937·41-s + 0.609·43-s − 0.583·47-s + 1.37·53-s + 0.256·61-s − 0.124·65-s + 0.977·67-s − 0.474·71-s + 0.702·73-s − 0.900·79-s + 0.878·83-s + 0.216·85-s − 0.635·89-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.11765221303536, −12.79279992482162, −12.18142349576765, −11.81687379408235, −11.36769456776898, −10.80511043636849, −10.23833381329523, −9.983309021307615, −9.495965907212857, −8.966975743832673, −8.467928406526003, −7.987602902129988, −7.526768298916818, −6.904200852489066, −6.519387289047037, −5.951347710813214, −5.489857581739772, −4.937257090286985, −4.500415072706527, −3.797263825585870, −3.333319111265613, −2.507458045243975, −2.340515768216701, −1.360409075124873, −0.9306700426702671, 0,
0.9306700426702671, 1.360409075124873, 2.340515768216701, 2.507458045243975, 3.333319111265613, 3.797263825585870, 4.500415072706527, 4.937257090286985, 5.489857581739772, 5.951347710813214, 6.519387289047037, 6.904200852489066, 7.526768298916818, 7.987602902129988, 8.467928406526003, 8.966975743832673, 9.495965907212857, 9.983309021307615, 10.23833381329523, 10.80511043636849, 11.36769456776898, 11.81687379408235, 12.18142349576765, 12.79279992482162, 13.11765221303536
