| L(s) = 1 | + 3-s − 7-s + 9-s + 11-s − 13-s − 17-s + 19-s − 21-s − 23-s + 27-s − 29-s − 31-s + 33-s + 37-s − 39-s − 41-s − 43-s + 47-s + 49-s − 51-s − 53-s + 57-s + 59-s + 61-s − 63-s + 67-s − 69-s + ⋯ |
| L(s) = 1 | + 3-s − 7-s + 9-s + 11-s − 13-s − 17-s + 19-s − 21-s − 23-s + 27-s − 29-s − 31-s + 33-s + 37-s − 39-s − 41-s − 43-s + 47-s + 49-s − 51-s − 53-s + 57-s + 59-s + 61-s − 63-s + 67-s − 69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.191102349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.191102349\) |
| \(L(1)\) |
\(\approx\) |
\(1.330350884\) |
| \(L(1)\) |
\(\approx\) |
\(1.330350884\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 151 | \( 1 \) |
| good | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 - T \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.84572546340777587613990356960, −16.82151752200028970901598284738, −16.45800763417807595350529903665, −15.55644844800410273405655687172, −15.15675622454606226908579899250, −14.31907906064263717838980624009, −13.90316048897083100461625380436, −13.109997562275053236668068107013, −12.61934176512967081005604073518, −11.87858596298613056300211508300, −11.139697357710059980481912287879, −10.05277742126011672123183773990, −9.6180125596525864794964090913, −9.214875938915944235730284693928, −8.4322758040825870335688198854, −7.59367819395416013117498662941, −6.97752424095057741160345406104, −6.467046326993480681520865369951, −5.48214366092040450166432172800, −4.5739512682625559490687409647, −3.726175844124176374889646933, −3.408257703000843959789318706092, −2.33797013616054143901572467959, −1.90475308835085493577556889317, −0.67227927420350327707006009979,
0.67227927420350327707006009979, 1.90475308835085493577556889317, 2.33797013616054143901572467959, 3.408257703000843959789318706092, 3.726175844124176374889646933, 4.5739512682625559490687409647, 5.48214366092040450166432172800, 6.467046326993480681520865369951, 6.97752424095057741160345406104, 7.59367819395416013117498662941, 8.4322758040825870335688198854, 9.214875938915944235730284693928, 9.6180125596525864794964090913, 10.05277742126011672123183773990, 11.139697357710059980481912287879, 11.87858596298613056300211508300, 12.61934176512967081005604073518, 13.109997562275053236668068107013, 13.90316048897083100461625380436, 14.31907906064263717838980624009, 15.15675622454606226908579899250, 15.55644844800410273405655687172, 16.45800763417807595350529903665, 16.82151752200028970901598284738, 17.84572546340777587613990356960
