| L(s) = 1 | − 3-s + 5-s + 9-s − 11-s − 13-s − 15-s − 17-s − 19-s + 23-s + 25-s − 27-s − 29-s − 31-s + 33-s + 37-s + 39-s − 41-s − 43-s + 45-s − 47-s + 51-s − 53-s − 55-s + 57-s − 61-s − 65-s − 67-s + ⋯ |
| L(s) = 1 | − 3-s + 5-s + 9-s − 11-s − 13-s − 15-s − 17-s − 19-s + 23-s + 25-s − 27-s − 29-s − 31-s + 33-s + 37-s + 39-s − 41-s − 43-s + 45-s − 47-s + 51-s − 53-s − 55-s + 57-s − 61-s − 65-s − 67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5656851888\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5656851888\) |
| \(L(1)\) |
\(\approx\) |
\(0.6558599316\) |
| \(L(1)\) |
\(\approx\) |
\(0.6558599316\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 59 | \( 1 \) |
| good | 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 | \( 1 - T \) |
| 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
−18.43805797646967578001740406194, −17.99993146613127861553572359869, −17.203090828228872293311200243953, −16.85041352972892129224413265264, −16.13578817808302855278301255319, −15.06766579021270480174009639911, −14.823188987468303047365720002299, −13.4944149818950977757264483087, −13.034021870003032470434479405446, −12.62834153228909666720441354378, −11.56479836522476904988749539058, −10.85192358115556493106420292276, −10.39544192445190206603386521869, −9.606258489284407526455872243688, −8.98517539630324028165032824752, −7.863033469571750014086724297, −7.03607121111657445977568763785, −6.44313171832481202901438369547, −5.665125205784439118474205893264, −4.9627864141946468073169033361, −4.53352862653458423455468225639, −3.17453737489010257783083624862, −2.19195477553284832580549757151, −1.62551903928287374975659771503, −0.28058324091586624129056689544,
0.28058324091586624129056689544, 1.62551903928287374975659771503, 2.19195477553284832580549757151, 3.17453737489010257783083624862, 4.53352862653458423455468225639, 4.9627864141946468073169033361, 5.665125205784439118474205893264, 6.44313171832481202901438369547, 7.03607121111657445977568763785, 7.863033469571750014086724297, 8.98517539630324028165032824752, 9.606258489284407526455872243688, 10.39544192445190206603386521869, 10.85192358115556493106420292276, 11.56479836522476904988749539058, 12.62834153228909666720441354378, 13.034021870003032470434479405446, 13.4944149818950977757264483087, 14.823188987468303047365720002299, 15.06766579021270480174009639911, 16.13578817808302855278301255319, 16.85041352972892129224413265264, 17.203090828228872293311200243953, 17.99993146613127861553572359869, 18.43805797646967578001740406194
