| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 11-s − 13-s + 14-s + 16-s − 17-s − 19-s − 22-s + 23-s + 26-s − 28-s − 31-s − 32-s + 34-s + 37-s + 38-s + 41-s + 43-s + 44-s − 46-s − 47-s + 49-s − 52-s + 53-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 11-s − 13-s + 14-s + 16-s − 17-s − 19-s − 22-s + 23-s + 26-s − 28-s − 31-s − 32-s + 34-s + 37-s + 38-s + 41-s + 43-s + 44-s − 46-s − 47-s + 49-s − 52-s + 53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8003944997\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8003944997\) |
| \(L(1)\) |
\(\approx\) |
\(0.6025113542\) |
| \(L(1)\) |
\(\approx\) |
\(0.6025113542\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 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 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−24.21649630334711204388469463636, −23.003324046523983853431477828039, −22.08106414818492329034188288645, −21.28192572272282020737363789878, −19.94334377550588716954757647773, −19.63137004600196349229114578799, −18.85480796964353907718003542737, −17.7451565396487420453392746699, −16.95047119341094338043369325699, −16.36973059379572984203898470118, −15.25399777408921672965982575732, −14.59061756119628453276042821863, −13.075394759368819861643578138502, −12.32381255292160690095268941761, −11.27088766298430001199390233692, −10.41292540062158994930253878483, −9.28872845580465017941777112707, −8.99870619189365054818276542947, −7.542253840796251426452541319131, −6.74194501749104339493084718755, −5.992996704537021704968202805847, −4.351663874297785056657980955028, −3.04680955804736040464119932757, −2.02723499398324805624239895303, −0.55520049399039061998617534658,
0.55520049399039061998617534658, 2.02723499398324805624239895303, 3.04680955804736040464119932757, 4.351663874297785056657980955028, 5.992996704537021704968202805847, 6.74194501749104339493084718755, 7.542253840796251426452541319131, 8.99870619189365054818276542947, 9.28872845580465017941777112707, 10.41292540062158994930253878483, 11.27088766298430001199390233692, 12.32381255292160690095268941761, 13.075394759368819861643578138502, 14.59061756119628453276042821863, 15.25399777408921672965982575732, 16.36973059379572984203898470118, 16.95047119341094338043369325699, 17.7451565396487420453392746699, 18.85480796964353907718003542737, 19.63137004600196349229114578799, 19.94334377550588716954757647773, 21.28192572272282020737363789878, 22.08106414818492329034188288645, 23.003324046523983853431477828039, 24.21649630334711204388469463636
