L(s) = 1 | + 1.24·2-s − 3-s − 0.445·4-s + 4.06·5-s − 1.24·6-s − 7-s − 3.04·8-s + 9-s + 5.07·10-s − 0.456·11-s + 0.445·12-s − 1.24·14-s − 4.06·15-s − 2.91·16-s − 5.72·17-s + 1.24·18-s + 0.188·19-s − 1.81·20-s + 21-s − 0.569·22-s − 5.05·23-s + 3.04·24-s + 11.5·25-s − 27-s + 0.445·28-s − 6.16·29-s − 5.07·30-s + ⋯ |
L(s) = 1 | + 0.881·2-s − 0.577·3-s − 0.222·4-s + 1.82·5-s − 0.509·6-s − 0.377·7-s − 1.07·8-s + 0.333·9-s + 1.60·10-s − 0.137·11-s + 0.128·12-s − 0.333·14-s − 1.05·15-s − 0.727·16-s − 1.38·17-s + 0.293·18-s + 0.0432·19-s − 0.405·20-s + 0.218·21-s − 0.121·22-s − 1.05·23-s + 0.622·24-s + 2.31·25-s − 0.192·27-s + 0.0841·28-s − 1.14·29-s − 0.926·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 1.24T + 2T^{2} \) |
| 5 | \( 1 - 4.06T + 5T^{2} \) |
| 11 | \( 1 + 0.456T + 11T^{2} \) |
| 17 | \( 1 + 5.72T + 17T^{2} \) |
| 19 | \( 1 - 0.188T + 19T^{2} \) |
| 23 | \( 1 + 5.05T + 23T^{2} \) |
| 29 | \( 1 + 6.16T + 29T^{2} \) |
| 31 | \( 1 - 1.19T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 + 2.45T + 41T^{2} \) |
| 43 | \( 1 - 4.28T + 43T^{2} \) |
| 47 | \( 1 + 7.89T + 47T^{2} \) |
| 53 | \( 1 - 5.67T + 53T^{2} \) |
| 59 | \( 1 + 13.4T + 59T^{2} \) |
| 61 | \( 1 + 0.497T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 - 7.94T + 73T^{2} \) |
| 79 | \( 1 - 4.07T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 - 15.7T + 89T^{2} \) |
| 97 | \( 1 + 9.85T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.368158868647085385283519093579, −6.97130388933312932882338225430, −6.40322532043637264177882285391, −5.81931819343526976400100885885, −5.27396901164943975372431744144, −4.56708094070636324918409868809, −3.60506058322161640535281476863, −2.51832252966962291920432052055, −1.71267112218039215527183958272, 0,
1.71267112218039215527183958272, 2.51832252966962291920432052055, 3.60506058322161640535281476863, 4.56708094070636324918409868809, 5.27396901164943975372431744144, 5.81931819343526976400100885885, 6.40322532043637264177882285391, 6.97130388933312932882338225430, 8.368158868647085385283519093579