L(s) = 1 | + 5-s + 7-s + 13-s + 17-s − 19-s + 23-s + 25-s − 29-s − 31-s + 35-s − 37-s + 41-s − 43-s + 47-s + 49-s + 53-s − 59-s + 61-s + 65-s + 67-s + 71-s − 73-s + 79-s + 83-s + 85-s − 89-s + 91-s + ⋯ |
L(s) = 1 | + 5-s + 7-s + 13-s + 17-s − 19-s + 23-s + 25-s − 29-s − 31-s + 35-s − 37-s + 41-s − 43-s + 47-s + 49-s + 53-s − 59-s + 61-s + 65-s + 67-s + 71-s − 73-s + 79-s + 83-s + 85-s − 89-s + 91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.844548380\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.844548380\) |
\(L(1)\) |
\(\approx\) |
\(1.546813295\) |
\(L(1)\) |
\(\approx\) |
\(1.546813295\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 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 | \( 1 + T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−25.531960448537157631594276839459, −24.829157847397313069835967068313, −23.79704343202651028015311554008, −22.96427397529695889492662649037, −21.70549995786776940028664026649, −21.058147923555650251209338233908, −20.45387603207282371794004464792, −18.91959095759050338527810879680, −18.23150594034769273386957970013, −17.265927884604002724014016745818, −16.58327959601650045146752714511, −15.139373570533744946321155565089, −14.3742580244394296721147910294, −13.4640543843162528218905202283, −12.524746174620956357801102927243, −11.167279887507364811712729873020, −10.50622664166826051946269658673, −9.22851836057645568838227204128, −8.38920597741791808642770967390, −7.13580286638191258357763543918, −5.88686609056203021622173216366, −5.097855274008953324543337648500, −3.691154311760920389665436180617, −2.16453672480886159225963517091, −1.16648103238616476814008084455,
1.16648103238616476814008084455, 2.16453672480886159225963517091, 3.691154311760920389665436180617, 5.097855274008953324543337648500, 5.88686609056203021622173216366, 7.13580286638191258357763543918, 8.38920597741791808642770967390, 9.22851836057645568838227204128, 10.50622664166826051946269658673, 11.167279887507364811712729873020, 12.524746174620956357801102927243, 13.4640543843162528218905202283, 14.3742580244394296721147910294, 15.139373570533744946321155565089, 16.58327959601650045146752714511, 17.265927884604002724014016745818, 18.23150594034769273386957970013, 18.91959095759050338527810879680, 20.45387603207282371794004464792, 21.058147923555650251209338233908, 21.70549995786776940028664026649, 22.96427397529695889492662649037, 23.79704343202651028015311554008, 24.829157847397313069835967068313, 25.531960448537157631594276839459