| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s + 11-s + 12-s + 13-s − 15-s + 16-s + 17-s + 18-s + 19-s − 20-s + 22-s − 23-s + 24-s + 25-s + 26-s + 27-s + 29-s − 30-s + 31-s + 32-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s + 11-s + 12-s + 13-s − 15-s + 16-s + 17-s + 18-s + 19-s − 20-s + 22-s − 23-s + 24-s + 25-s + 26-s + 27-s + 29-s − 30-s + 31-s + 32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2807 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2807 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(8.781271427\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.781271427\) |
| \(L(1)\) |
\(\approx\) |
\(3.083415207\) |
| \(L(1)\) |
\(\approx\) |
\(3.083415207\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 401 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 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
−19.325856248201720626364078041420, −18.69478350566394495528655984700, −17.63181008234720432132500831816, −16.454156593822760163885254697909, −15.90161610932003912682774910018, −15.550859589025267146930486834971, −14.51633929104803501739299738012, −14.23100840563775232083285392432, −13.547064464983647296087969343879, −12.67855546892235404619029098003, −11.884856908287531807423289312216, −11.62099656876368509493274795177, −10.44235895343634875174438057730, −9.81721959361931359531663149202, −8.619334685148597786923401650299, −8.15166061528971826457413954154, −7.33083164645182702762915953151, −6.70519836668810131781554936804, −5.813938505748765599058498866147, −4.68674927660028702509782212036, −4.05386295233828382524939291778, −3.38618683579592075914691739048, −2.95868520446577756763486041100, −1.61932950252551192134800978067, −1.01460465449165632913833435486,
1.01460465449165632913833435486, 1.61932950252551192134800978067, 2.95868520446577756763486041100, 3.38618683579592075914691739048, 4.05386295233828382524939291778, 4.68674927660028702509782212036, 5.813938505748765599058498866147, 6.70519836668810131781554936804, 7.33083164645182702762915953151, 8.15166061528971826457413954154, 8.619334685148597786923401650299, 9.81721959361931359531663149202, 10.44235895343634875174438057730, 11.62099656876368509493274795177, 11.884856908287531807423289312216, 12.67855546892235404619029098003, 13.547064464983647296087969343879, 14.23100840563775232083285392432, 14.51633929104803501739299738012, 15.550859589025267146930486834971, 15.90161610932003912682774910018, 16.454156593822760163885254697909, 17.63181008234720432132500831816, 18.69478350566394495528655984700, 19.325856248201720626364078041420
