L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 3·7-s − 8-s + 9-s + 1.60·11-s − 12-s + 5.60·13-s − 3·14-s + 16-s + 2·17-s − 18-s + 2·19-s − 3·21-s − 1.60·22-s + 7.21·23-s + 24-s − 5.60·26-s − 27-s + 3·28-s + 7.21·29-s − 31-s − 32-s − 1.60·33-s − 2·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s + 1.13·7-s − 0.353·8-s + 0.333·9-s + 0.484·11-s − 0.288·12-s + 1.55·13-s − 0.801·14-s + 0.250·16-s + 0.485·17-s − 0.235·18-s + 0.458·19-s − 0.654·21-s − 0.342·22-s + 1.50·23-s + 0.204·24-s − 1.09·26-s − 0.192·27-s + 0.566·28-s + 1.33·29-s − 0.179·31-s − 0.176·32-s − 0.279·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.755039858\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.755039858\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 31 | \( 1 + T \) |
good | 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 - 1.60T + 11T^{2} \) |
| 13 | \( 1 - 5.60T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 7.21T + 23T^{2} \) |
| 29 | \( 1 - 7.21T + 29T^{2} \) |
| 37 | \( 1 - 9.60T + 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 - 2.39T + 43T^{2} \) |
| 47 | \( 1 - 3T + 47T^{2} \) |
| 53 | \( 1 + 8.81T + 53T^{2} \) |
| 59 | \( 1 + 3.21T + 59T^{2} \) |
| 61 | \( 1 + 5.60T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 9T + 71T^{2} \) |
| 73 | \( 1 - 16.4T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 - 6.39T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−8.160618978527970766256644478601, −7.84862680551658749915502736127, −6.79901365156119077680335391174, −6.29608192347298579551309426382, −5.42594307882798300568386963199, −4.71920621830672004233394680275, −3.78261152804357199846682197163, −2.77057517285995213758129520381, −1.38623777371052048988454164608, −1.02810268830016698013118943652,
1.02810268830016698013118943652, 1.38623777371052048988454164608, 2.77057517285995213758129520381, 3.78261152804357199846682197163, 4.71920621830672004233394680275, 5.42594307882798300568386963199, 6.29608192347298579551309426382, 6.79901365156119077680335391174, 7.84862680551658749915502736127, 8.160618978527970766256644478601