| L(s) = 1 | + 3-s + 5-s + 7-s + 9-s − 11-s + 13-s + 15-s + 17-s − 19-s + 21-s − 23-s + 25-s + 27-s + 29-s − 31-s − 33-s + 35-s − 37-s + 39-s − 41-s + 43-s + 45-s − 47-s + 49-s + 51-s + 53-s − 55-s + ⋯ |
| L(s) = 1 | + 3-s + 5-s + 7-s + 9-s − 11-s + 13-s + 15-s + 17-s − 19-s + 21-s − 23-s + 25-s + 27-s + 29-s − 31-s − 33-s + 35-s − 37-s + 39-s − 41-s + 43-s + 45-s − 47-s + 49-s + 51-s + 53-s − 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.024832672\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.024832672\) |
| \(L(1)\) |
\(\approx\) |
\(1.919262630\) |
| \(L(1)\) |
\(\approx\) |
\(1.919262630\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 137 | \( 1 \) |
| good | 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
−21.2742859119788537711418750422, −20.79506424312550062907126135751, −20.111260106430172465447799552025, −18.94798253456668246641684098874, −18.30096970724017081601947465589, −17.82051137067902195263950908148, −16.7468929888669167613016044680, −15.84391062424642672340213699154, −15.020596194843729573557349575170, −14.21478500105300713083926477688, −13.7605511666619484189130175551, −12.973207063328405446467343047091, −12.13631284945281289181150338306, −10.69693500388486442046686332346, −10.38516375232313060869646908466, −9.35502392810757715616030918537, −8.416166365069968638894466214803, −8.04253626275362176551504588165, −6.93452596122917941510508854099, −5.85286722878375633112590763034, −5.04010739419190630055605171840, −4.01724937471545500503814695272, −2.952862855198216013087520060536, −2.021623916296631161428008943794, −1.37897092512164650154404840041,
1.37897092512164650154404840041, 2.021623916296631161428008943794, 2.952862855198216013087520060536, 4.01724937471545500503814695272, 5.04010739419190630055605171840, 5.85286722878375633112590763034, 6.93452596122917941510508854099, 8.04253626275362176551504588165, 8.416166365069968638894466214803, 9.35502392810757715616030918537, 10.38516375232313060869646908466, 10.69693500388486442046686332346, 12.13631284945281289181150338306, 12.973207063328405446467343047091, 13.7605511666619484189130175551, 14.21478500105300713083926477688, 15.020596194843729573557349575170, 15.84391062424642672340213699154, 16.7468929888669167613016044680, 17.82051137067902195263950908148, 18.30096970724017081601947465589, 18.94798253456668246641684098874, 20.111260106430172465447799552025, 20.79506424312550062907126135751, 21.2742859119788537711418750422
