L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s + 12-s + 13-s − 14-s + 16-s − 17-s − 18-s + 19-s + 21-s − 23-s − 24-s − 26-s + 27-s + 28-s − 31-s − 32-s + 34-s + 36-s + 37-s − 38-s + 39-s + ⋯ |
L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s + 12-s + 13-s − 14-s + 16-s − 17-s − 18-s + 19-s + 21-s − 23-s − 24-s − 26-s + 27-s + 28-s − 31-s − 32-s + 34-s + 36-s + 37-s − 38-s + 39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1595 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1595 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.851061220\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.851061220\) |
\(L(1)\) |
\(\approx\) |
\(1.258605170\) |
\(L(1)\) |
\(\approx\) |
\(1.258605170\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 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
−20.32422738761110806734669080721, −19.64004088711262467258956055577, −18.67308606257744239504878648967, −18.08906117295005369423455951417, −17.709032304251433165554449315541, −16.43964445238081412037201547656, −15.870537167292016032241162754495, −15.154040131675119057188572427196, −14.4084705399220922932724858918, −13.6760249511875549062490229146, −12.762078853615363375901181934954, −11.67696136259021555456648870015, −11.07383083568319737663367949815, −10.2633834218841575034563967289, −9.34478356965383702000704775800, −8.77472357181005596264584211620, −8.03434397515108111932722494276, −7.51942816340483804057279642601, −6.58087361508960945180807414880, −5.56849676743241641329640243474, −4.31501041231563595365235240021, −3.45603867540817036241700955044, −2.37211369468768045764538003587, −1.71193246749524997277366733146, −0.82372923073299959559515880700,
0.82372923073299959559515880700, 1.71193246749524997277366733146, 2.37211369468768045764538003587, 3.45603867540817036241700955044, 4.31501041231563595365235240021, 5.56849676743241641329640243474, 6.58087361508960945180807414880, 7.51942816340483804057279642601, 8.03434397515108111932722494276, 8.77472357181005596264584211620, 9.34478356965383702000704775800, 10.2633834218841575034563967289, 11.07383083568319737663367949815, 11.67696136259021555456648870015, 12.762078853615363375901181934954, 13.6760249511875549062490229146, 14.4084705399220922932724858918, 15.154040131675119057188572427196, 15.870537167292016032241162754495, 16.43964445238081412037201547656, 17.709032304251433165554449315541, 18.08906117295005369423455951417, 18.67308606257744239504878648967, 19.64004088711262467258956055577, 20.32422738761110806734669080721