L(s) = 1 | − 3-s + 4.96·7-s + 9-s + 0.430·11-s + 13-s − 3·17-s + 3.39·19-s − 4.96·21-s + 5.39·23-s − 27-s − 4.13·29-s + 0.430·31-s − 0.430·33-s + 8.53·37-s − 39-s − 6.53·41-s + 6.53·43-s − 12.1·47-s + 17.6·49-s + 3·51-s + 4.39·53-s − 3.39·57-s + 12.1·59-s + 6.13·61-s + 4.96·63-s − 9.56·67-s − 5.39·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.87·7-s + 0.333·9-s + 0.129·11-s + 0.277·13-s − 0.727·17-s + 0.779·19-s − 1.08·21-s + 1.12·23-s − 0.192·27-s − 0.768·29-s + 0.0773·31-s − 0.0749·33-s + 1.40·37-s − 0.160·39-s − 1.02·41-s + 0.996·43-s − 1.76·47-s + 2.52·49-s + 0.420·51-s + 0.604·53-s − 0.450·57-s + 1.57·59-s + 0.785·61-s + 0.625·63-s − 1.16·67-s − 0.649·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.371597264\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.371597264\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 - 4.96T + 7T^{2} \) |
| 11 | \( 1 - 0.430T + 11T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - 3.39T + 19T^{2} \) |
| 23 | \( 1 - 5.39T + 23T^{2} \) |
| 29 | \( 1 + 4.13T + 29T^{2} \) |
| 31 | \( 1 - 0.430T + 31T^{2} \) |
| 37 | \( 1 - 8.53T + 37T^{2} \) |
| 41 | \( 1 + 6.53T + 41T^{2} \) |
| 43 | \( 1 - 6.53T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 - 4.39T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 - 6.13T + 61T^{2} \) |
| 67 | \( 1 + 9.56T + 67T^{2} \) |
| 71 | \( 1 - 9.67T + 71T^{2} \) |
| 73 | \( 1 + 9.67T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 - 7.82T + 83T^{2} \) |
| 89 | \( 1 - 2.86T + 89T^{2} \) |
| 97 | \( 1 - 5.93T + 97T^{2} \) |
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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
−7.83546194131272974913739400765, −7.20838727495613601108673719250, −6.50632224340359983542609703050, −5.56015353867194372565392774434, −5.08261257586526528746151853632, −4.49441171111645072918124767353, −3.71395206035176505803930664235, −2.50657134113328661151702658932, −1.62145352389692138147839197307, −0.857419905429107983260192090025,
0.857419905429107983260192090025, 1.62145352389692138147839197307, 2.50657134113328661151702658932, 3.71395206035176505803930664235, 4.49441171111645072918124767353, 5.08261257586526528746151853632, 5.56015353867194372565392774434, 6.50632224340359983542609703050, 7.20838727495613601108673719250, 7.83546194131272974913739400765