| L(s) = 1 | + 7-s + 4·13-s + 2·17-s + 8·29-s + 4·31-s + 8·37-s + 4·41-s − 8·43-s + 12·47-s + 49-s + 6·53-s − 8·59-s + 10·61-s − 8·67-s + 16·71-s + 12·73-s + 8·79-s − 16·83-s − 12·89-s + 4·91-s + 4·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 2·119-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1.10·13-s + 0.485·17-s + 1.48·29-s + 0.718·31-s + 1.31·37-s + 0.624·41-s − 1.21·43-s + 1.75·47-s + 1/7·49-s + 0.824·53-s − 1.04·59-s + 1.28·61-s − 0.977·67-s + 1.89·71-s + 1.40·73-s + 0.900·79-s − 1.75·83-s − 1.27·89-s + 0.419·91-s + 0.406·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.183·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.072476069\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.072476069\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−15.46904189441893, −14.85472498330776, −14.15091959722639, −13.83243215527963, −13.31998854333400, −12.61167654692280, −12.13525653541977, −11.55303801165129, −11.03413476254441, −10.49293132653804, −9.953098908299663, −9.303445916619740, −8.666624743258326, −8.129498602998694, −7.766453318073139, −6.807593749297405, −6.434048275242280, −5.687612203079820, −5.168268100940868, −4.307356843695210, −3.900943523971011, −2.975857046144583, −2.400322806072720, −1.324894881504977, −0.7846234187634006,
0.7846234187634006, 1.324894881504977, 2.400322806072720, 2.975857046144583, 3.900943523971011, 4.307356843695210, 5.168268100940868, 5.687612203079820, 6.434048275242280, 6.807593749297405, 7.766453318073139, 8.129498602998694, 8.666624743258326, 9.303445916619740, 9.953098908299663, 10.49293132653804, 11.03413476254441, 11.55303801165129, 12.13525653541977, 12.61167654692280, 13.31998854333400, 13.83243215527963, 14.15091959722639, 14.85472498330776, 15.46904189441893
