L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 1.07·7-s + 8-s + 9-s + 3.41·11-s + 12-s + 0.921·13-s + 1.07·14-s + 16-s − 0.340·17-s + 18-s − 19-s + 1.07·21-s + 3.41·22-s + 0.921·23-s + 24-s + 0.921·26-s + 27-s + 1.07·28-s − 1.41·29-s + 7.26·31-s + 32-s + 3.41·33-s − 0.340·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s + 0.407·7-s + 0.353·8-s + 0.333·9-s + 1.03·11-s + 0.288·12-s + 0.255·13-s + 0.288·14-s + 0.250·16-s − 0.0825·17-s + 0.235·18-s − 0.229·19-s + 0.235·21-s + 0.728·22-s + 0.192·23-s + 0.204·24-s + 0.180·26-s + 0.192·27-s + 0.203·28-s − 0.263·29-s + 1.30·31-s + 0.176·32-s + 0.595·33-s − 0.0583·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.174608576\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.174608576\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 - 1.07T + 7T^{2} \) |
| 11 | \( 1 - 3.41T + 11T^{2} \) |
| 13 | \( 1 - 0.921T + 13T^{2} \) |
| 17 | \( 1 + 0.340T + 17T^{2} \) |
| 23 | \( 1 - 0.921T + 23T^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 31 | \( 1 - 7.26T + 31T^{2} \) |
| 37 | \( 1 + 5.60T + 37T^{2} \) |
| 41 | \( 1 + 1.07T + 41T^{2} \) |
| 43 | \( 1 + 0.738T + 43T^{2} \) |
| 47 | \( 1 - 7.75T + 47T^{2} \) |
| 53 | \( 1 + 2.68T + 53T^{2} \) |
| 59 | \( 1 - 8.34T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 4.68T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 - 6.83T + 73T^{2} \) |
| 79 | \( 1 - 4.73T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 - 9.75T + 89T^{2} \) |
| 97 | \( 1 + 16.2T + 97T^{2} \) |
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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
−8.660155942852118517427050910785, −8.097462488799357899480993757050, −7.11150461525380051591678627816, −6.56115326544906252245241438575, −5.66936420649730443435582694185, −4.71688758940692180821682921486, −4.02769365733197114508109859252, −3.25134054008919156790089455244, −2.21896005726333935424128097692, −1.23499742185961202874085860454,
1.23499742185961202874085860454, 2.21896005726333935424128097692, 3.25134054008919156790089455244, 4.02769365733197114508109859252, 4.71688758940692180821682921486, 5.66936420649730443435582694185, 6.56115326544906252245241438575, 7.11150461525380051591678627816, 8.097462488799357899480993757050, 8.660155942852118517427050910785