L(s) = 1 | − 1.87·2-s − 1.53·3-s + 2.53·4-s + 1.87·5-s + 2.87·6-s + 7-s − 2.87·8-s + 1.34·9-s − 3.53·10-s + 11-s − 3.87·12-s + 13-s − 1.87·14-s − 2.87·15-s + 2.87·16-s − 0.347·17-s − 2.53·18-s + 4.75·20-s − 1.53·21-s − 1.87·22-s − 23-s + 4.41·24-s + 2.53·25-s − 1.87·26-s − 0.532·27-s + 2.53·28-s + 1.53·29-s + ⋯ |
L(s) = 1 | − 1.87·2-s − 1.53·3-s + 2.53·4-s + 1.87·5-s + 2.87·6-s + 7-s − 2.87·8-s + 1.34·9-s − 3.53·10-s + 11-s − 3.87·12-s + 13-s − 1.87·14-s − 2.87·15-s + 2.87·16-s − 0.347·17-s − 2.53·18-s + 4.75·20-s − 1.53·21-s − 1.87·22-s − 23-s + 4.41·24-s + 2.53·25-s − 1.87·26-s − 0.532·27-s + 2.53·28-s + 1.53·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6118737738\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6118737738\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 - T \) |
good | 2 | \( 1 + 1.87T + T^{2} \) |
| 3 | \( 1 + 1.53T + T^{2} \) |
| 5 | \( 1 - 1.87T + T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 + 0.347T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 - 1.53T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 0.347T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 0.347T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.87T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.53T + T^{2} \) |
| 89 | \( 1 + 2T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.820907628785638015877775555640, −8.425645043108580951612449419865, −7.19426177888150238745669960175, −6.52048485066540500825120694662, −6.06302169485209141663847961218, −5.55814748472375137751210012237, −4.42399214841441578022954060346, −2.56070836490988475395305869405, −1.50771114210261946106002010694, −1.18542852239258822122214858277,
1.18542852239258822122214858277, 1.50771114210261946106002010694, 2.56070836490988475395305869405, 4.42399214841441578022954060346, 5.55814748472375137751210012237, 6.06302169485209141663847961218, 6.52048485066540500825120694662, 7.19426177888150238745669960175, 8.425645043108580951612449419865, 8.820907628785638015877775555640