| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 7-s + 8-s + 9-s − 10-s + 11-s − 12-s − 13-s + 14-s + 15-s + 16-s + 17-s + 18-s + 19-s − 20-s − 21-s + 22-s − 23-s − 24-s + 25-s − 26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 7-s + 8-s + 9-s − 10-s + 11-s − 12-s − 13-s + 14-s + 15-s + 16-s + 17-s + 18-s + 19-s − 20-s − 21-s + 22-s − 23-s − 24-s + 25-s − 26-s − 27-s + 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.467828239\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.467828239\) |
| \(L(1)\) |
\(\approx\) |
\(1.393813413\) |
| \(L(1)\) |
\(\approx\) |
\(1.393813413\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 137 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 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
−28.490650194071507166929583197350, −27.62597752839115843173861899911, −26.820136069382704049458334274897, −24.99247551136163590094809581519, −24.09852245786282625734384366359, −23.62325592495157004807575073467, −22.42517123792418293698526841276, −21.93474075278556387533945551433, −20.61849057876589501302475842413, −19.67446596285060651700157907754, −18.36853308286686107381698992475, −16.96928070760936581370901930058, −16.290203368438110587379663978804, −15.0230266176896080859872935267, −14.33052308016035524420243833288, −12.67978812911380365358728946894, −11.64402320158849251907815308237, −11.51994282544541089398096360070, −9.99487934200130908187822998186, −7.82253738263475542233360177680, −7.05007495939429706747243604079, −5.58151662785808071183597329855, −4.66054959833737303952446070246, −3.632768617007316789685039196525, −1.52794551399684727386219317429,
1.52794551399684727386219317429, 3.632768617007316789685039196525, 4.66054959833737303952446070246, 5.58151662785808071183597329855, 7.05007495939429706747243604079, 7.82253738263475542233360177680, 9.99487934200130908187822998186, 11.51994282544541089398096360070, 11.64402320158849251907815308237, 12.67978812911380365358728946894, 14.33052308016035524420243833288, 15.0230266176896080859872935267, 16.290203368438110587379663978804, 16.96928070760936581370901930058, 18.36853308286686107381698992475, 19.67446596285060651700157907754, 20.61849057876589501302475842413, 21.93474075278556387533945551433, 22.42517123792418293698526841276, 23.62325592495157004807575073467, 24.09852245786282625734384366359, 24.99247551136163590094809581519, 26.820136069382704049458334274897, 27.62597752839115843173861899911, 28.490650194071507166929583197350
