| 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 + 24-s + 25-s − 26-s − 27-s − 28-s − 30-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 + 24-s + 25-s − 26-s − 27-s − 28-s − 30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6988598802\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6988598802\) |
| \(L(1)\) |
\(\approx\) |
\(0.4865718133\) |
| \(L(1)\) |
\(\approx\) |
\(0.4865718133\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 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
−22.66116104468945981854579215835, −21.90973131689210430893921635657, −20.71444343027866082208030884826, −19.93667181166727446299697092530, −19.06456646888736374547119844733, −18.59666025623962465263097974542, −17.689469390690416651350041013154, −16.5496207517143350188763461226, −16.3345738943921906400932118490, −15.62528902093452673062456127362, −14.5918328601518102329922086955, −13.0325421120074152330811158904, −12.145774853701613855869099858307, −11.58380502389937507812421456782, −10.84632696346750916492131724003, −9.855001680231939791299673196105, −9.14127999489666318881059494881, −7.96333494861814155774235695751, −7.0882810835242355999052852798, −6.402846186525547927728926457130, −5.51225321720528189031212236380, −3.91390059049755598789300947720, −3.23257648409304430757695082965, −1.35363359080315757327540129387, −0.586080954139019066925136901801,
0.586080954139019066925136901801, 1.35363359080315757327540129387, 3.23257648409304430757695082965, 3.91390059049755598789300947720, 5.51225321720528189031212236380, 6.402846186525547927728926457130, 7.0882810835242355999052852798, 7.96333494861814155774235695751, 9.14127999489666318881059494881, 9.855001680231939791299673196105, 10.84632696346750916492131724003, 11.58380502389937507812421456782, 12.145774853701613855869099858307, 13.0325421120074152330811158904, 14.5918328601518102329922086955, 15.62528902093452673062456127362, 16.3345738943921906400932118490, 16.5496207517143350188763461226, 17.689469390690416651350041013154, 18.59666025623962465263097974542, 19.06456646888736374547119844733, 19.93667181166727446299697092530, 20.71444343027866082208030884826, 21.90973131689210430893921635657, 22.66116104468945981854579215835
