| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2636 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2636 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.717460377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.717460377\) |
| \(L(1)\) |
\(\approx\) |
\(1.096287060\) |
| \(L(1)\) |
\(\approx\) |
\(1.096287060\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 659 | \( 1 \) |
| 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
−18.84838363999957411470997221570, −18.533697041775964102506496104927, −17.89477415573060084425807484083, −17.250375611776590116934496365515, −16.73160636391125010412343057816, −15.87913306552958743086241735248, −15.18759706278738060935674328233, −14.26157918223953010474036838665, −13.593228548719019198632287534365, −12.87342224504382941539263743475, −12.21516217485509067800285463895, −11.26179635495306760055430247487, −10.726331049832516941704443917179, −10.17807040760187814655892589579, −9.37114005479962340189530918647, −8.25210956786193966218678299485, −7.744340420569234289976723533871, −6.61878686676864478090908377172, −5.900843827223446152060766898147, −5.43669423902397984988261388011, −4.669596235210806313559756091519, −3.80245296672508502662047134090, −2.4360374844084424669508425980, −1.70573782794460799748766949873, −0.85228753957497076816231170519,
0.85228753957497076816231170519, 1.70573782794460799748766949873, 2.4360374844084424669508425980, 3.80245296672508502662047134090, 4.669596235210806313559756091519, 5.43669423902397984988261388011, 5.900843827223446152060766898147, 6.61878686676864478090908377172, 7.744340420569234289976723533871, 8.25210956786193966218678299485, 9.37114005479962340189530918647, 10.17807040760187814655892589579, 10.726331049832516941704443917179, 11.26179635495306760055430247487, 12.21516217485509067800285463895, 12.87342224504382941539263743475, 13.593228548719019198632287534365, 14.26157918223953010474036838665, 15.18759706278738060935674328233, 15.87913306552958743086241735248, 16.73160636391125010412343057816, 17.250375611776590116934496365515, 17.89477415573060084425807484083, 18.533697041775964102506496104927, 18.84838363999957411470997221570
