| 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 + 29-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 + ⋯ |
| 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 + 29-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 915 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 915 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.502485849\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.502485849\) |
| \(L(1)\) |
\(\approx\) |
\(0.8308627957\) |
| \(L(1)\) |
\(\approx\) |
\(0.8308627957\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 61 | \( 1 \) |
| good | 2 | \( 1 - T \) |
| 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 | \( 1 + T \) |
| 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
−21.88747701156733947131509334682, −20.514091198315445785695201585361, −20.10454443235284857078474313381, −19.3895674973801592785290431169, −18.37051382159256829118688765395, −17.69404218749881731752412830174, −17.205834907490256822865693557604, −16.29019070467204559789517117618, −15.44800858088245977230532958259, −14.5769494021970679135224801728, −13.98460816568452744141407167655, −12.50583386417821603904005829140, −11.73944858475573174253795036879, −11.21293587407372581573785043495, −10.19136630112533482028006377619, −9.38864283529383548795238493151, −8.6413454681331202733070654505, −7.739398035189489556986981430692, −7.056806754791235581170233137078, −6.08313785554983175136294898810, −4.98600623201739455607424924295, −3.91005468719411936648766788298, −2.54565859463529656799956813083, −1.73375720416775968670569766990, −0.68862504180570466637659808733,
0.68862504180570466637659808733, 1.73375720416775968670569766990, 2.54565859463529656799956813083, 3.91005468719411936648766788298, 4.98600623201739455607424924295, 6.08313785554983175136294898810, 7.056806754791235581170233137078, 7.739398035189489556986981430692, 8.6413454681331202733070654505, 9.38864283529383548795238493151, 10.19136630112533482028006377619, 11.21293587407372581573785043495, 11.73944858475573174253795036879, 12.50583386417821603904005829140, 13.98460816568452744141407167655, 14.5769494021970679135224801728, 15.44800858088245977230532958259, 16.29019070467204559789517117618, 17.205834907490256822865693557604, 17.69404218749881731752412830174, 18.37051382159256829118688765395, 19.3895674973801592785290431169, 20.10454443235284857078474313381, 20.514091198315445785695201585361, 21.88747701156733947131509334682
