| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1399 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1399 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(5.578736460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.578736460\) |
| \(L(1)\) |
\(\approx\) |
\(2.267800149\) |
| \(L(1)\) |
\(\approx\) |
\(2.267800149\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1399 | \( 1 \) |
| 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 | \( 1 - T \) |
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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
−20.88163316534169674650715946537, −20.12217538622643303704126187795, −19.20524730573772107565820824040, −17.89302425741064377948215237462, −17.63127205258457838959076827056, −16.78061756846998736512708357312, −16.20914383121320699850381641095, −14.97617180074921225570205766801, −14.581309322857852141149792243743, −13.68668066047022202271479393525, −12.96115516440463794184539657171, −12.14763847107787463331852569386, −11.43139402279335930451521666609, −10.91751284387820118532592365435, −9.96211835945057848388249627324, −9.14290792615572654052269511865, −7.7037195599051119812337126322, −6.83516170679345997318470988045, −6.32378377976464882234043110456, −5.205634166021464708439504888038, −4.97341957221443798633811614104, −4.05537133574538480204041663168, −2.67879964637893118544871945785, −1.73415326426702909759362267623, −1.014272215499202888157972362335,
1.014272215499202888157972362335, 1.73415326426702909759362267623, 2.67879964637893118544871945785, 4.05537133574538480204041663168, 4.97341957221443798633811614104, 5.205634166021464708439504888038, 6.32378377976464882234043110456, 6.83516170679345997318470988045, 7.7037195599051119812337126322, 9.14290792615572654052269511865, 9.96211835945057848388249627324, 10.91751284387820118532592365435, 11.43139402279335930451521666609, 12.14763847107787463331852569386, 12.96115516440463794184539657171, 13.68668066047022202271479393525, 14.581309322857852141149792243743, 14.97617180074921225570205766801, 16.20914383121320699850381641095, 16.78061756846998736512708357312, 17.63127205258457838959076827056, 17.89302425741064377948215237462, 19.20524730573772107565820824040, 20.12217538622643303704126187795, 20.88163316534169674650715946537
