L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s + 11-s − 13-s − 14-s + 16-s + 17-s + 19-s − 20-s − 22-s + 23-s + 25-s + 26-s + 28-s + 29-s − 32-s − 34-s − 35-s − 37-s − 38-s + 40-s − 41-s − 43-s + ⋯ |
L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s + 11-s − 13-s − 14-s + 16-s + 17-s + 19-s − 20-s − 22-s + 23-s + 25-s + 26-s + 28-s + 29-s − 32-s − 34-s − 35-s − 37-s − 38-s + 40-s − 41-s − 43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6463882676\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6463882676\) |
\(L(1)\) |
\(\approx\) |
\(0.6980975828\) |
\(L(1)\) |
\(\approx\) |
\(0.6980975828\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| 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
−30.26894661363184977652819679892, −29.13905505248956128918760654205, −27.783752161004323503092243784889, −27.31812329870420989204006237363, −26.521252010624751365321100760965, −24.956377810014836349458855546862, −24.35425241125656561041315713588, −23.13389621357033109853129625168, −21.61437279880648113363784687092, −20.40030825270576800426378814414, −19.56797667291413676224705512310, −18.61641015162260692905051547214, −17.38514862043382879917342212004, −16.53293474971464328222143232570, −15.2183085794300512386783948356, −14.405059431081373406762477900138, −12.06254402016068766189096253464, −11.63856129187017707571303652051, −10.27376283236076307735704265953, −8.906088181404724898552890998775, −7.83426023162166228485446494924, −6.95142207703026649152112335629, −5.00169634180100983262293205910, −3.22956829113678628924654860399, −1.30110184040805982023272325805,
1.30110184040805982023272325805, 3.22956829113678628924654860399, 5.00169634180100983262293205910, 6.95142207703026649152112335629, 7.83426023162166228485446494924, 8.906088181404724898552890998775, 10.27376283236076307735704265953, 11.63856129187017707571303652051, 12.06254402016068766189096253464, 14.405059431081373406762477900138, 15.2183085794300512386783948356, 16.53293474971464328222143232570, 17.38514862043382879917342212004, 18.61641015162260692905051547214, 19.56797667291413676224705512310, 20.40030825270576800426378814414, 21.61437279880648113363784687092, 23.13389621357033109853129625168, 24.35425241125656561041315713588, 24.956377810014836349458855546862, 26.521252010624751365321100760965, 27.31812329870420989204006237363, 27.783752161004323503092243784889, 29.13905505248956128918760654205, 30.26894661363184977652819679892