| L(s) = 1 | + 4·11-s − 13-s − 8·17-s + 5·19-s − 4·23-s + 9·29-s + 4·31-s + 3·37-s − 5·41-s + 6·43-s − 5·47-s − 7·49-s − 5·53-s + 6·59-s − 4·61-s − 3·67-s + 7·71-s + 4·73-s + 79-s + 6·83-s + 6·89-s − 4·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.20·11-s − 0.277·13-s − 1.94·17-s + 1.14·19-s − 0.834·23-s + 1.67·29-s + 0.718·31-s + 0.493·37-s − 0.780·41-s + 0.914·43-s − 0.729·47-s − 49-s − 0.686·53-s + 0.781·59-s − 0.512·61-s − 0.366·67-s + 0.830·71-s + 0.468·73-s + 0.112·79-s + 0.658·83-s + 0.635·89-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.548931020\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.548931020\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.16268791507166, −12.60600544469531, −12.12325595634824, −11.60673960748697, −11.47055668326786, −10.79365211783698, −10.28242741686509, −9.698071510446492, −9.401574257416144, −8.887312839230879, −8.314247235952157, −8.010788909498234, −7.214579324580745, −6.819410389477187, −6.277506345709255, −6.105972067429034, −5.138988487736537, −4.630808745540791, −4.374640029405545, −3.616482627040172, −3.121005125009824, −2.399735429310370, −1.889313845639512, −1.141977350276584, −0.4894392569082353,
0.4894392569082353, 1.141977350276584, 1.889313845639512, 2.399735429310370, 3.121005125009824, 3.616482627040172, 4.374640029405545, 4.630808745540791, 5.138988487736537, 6.105972067429034, 6.277506345709255, 6.819410389477187, 7.214579324580745, 8.010788909498234, 8.314247235952157, 8.887312839230879, 9.401574257416144, 9.698071510446492, 10.28242741686509, 10.79365211783698, 11.47055668326786, 11.60673960748697, 12.12325595634824, 12.60600544469531, 13.16268791507166
