L(s) = 1 | − 2-s − 0.471·3-s + 4-s + 1.82·5-s + 0.471·6-s − 1.64·7-s − 8-s − 2.77·9-s − 1.82·10-s − 6.17·11-s − 0.471·12-s − 3.42·13-s + 1.64·14-s − 0.859·15-s + 16-s − 5.99·17-s + 2.77·18-s − 2.80·19-s + 1.82·20-s + 0.776·21-s + 6.17·22-s + 23-s + 0.471·24-s − 1.68·25-s + 3.42·26-s + 2.72·27-s − 1.64·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.272·3-s + 0.5·4-s + 0.814·5-s + 0.192·6-s − 0.621·7-s − 0.353·8-s − 0.925·9-s − 0.575·10-s − 1.86·11-s − 0.136·12-s − 0.950·13-s + 0.439·14-s − 0.221·15-s + 0.250·16-s − 1.45·17-s + 0.654·18-s − 0.644·19-s + 0.407·20-s + 0.169·21-s + 1.31·22-s + 0.208·23-s + 0.0963·24-s − 0.337·25-s + 0.671·26-s + 0.524·27-s − 0.310·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2268210475\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2268210475\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
good | 3 | \( 1 + 0.471T + 3T^{2} \) |
| 5 | \( 1 - 1.82T + 5T^{2} \) |
| 7 | \( 1 + 1.64T + 7T^{2} \) |
| 11 | \( 1 + 6.17T + 11T^{2} \) |
| 13 | \( 1 + 3.42T + 13T^{2} \) |
| 17 | \( 1 + 5.99T + 17T^{2} \) |
| 19 | \( 1 + 2.80T + 19T^{2} \) |
| 29 | \( 1 + 3.31T + 29T^{2} \) |
| 31 | \( 1 - 3.32T + 31T^{2} \) |
| 37 | \( 1 + 8.75T + 37T^{2} \) |
| 41 | \( 1 - 5.09T + 41T^{2} \) |
| 43 | \( 1 + 6.77T + 43T^{2} \) |
| 47 | \( 1 + 4.23T + 47T^{2} \) |
| 53 | \( 1 - 4.22T + 53T^{2} \) |
| 59 | \( 1 - 5.10T + 59T^{2} \) |
| 61 | \( 1 - 6.07T + 61T^{2} \) |
| 67 | \( 1 - 8.73T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 5.69T + 73T^{2} \) |
| 79 | \( 1 - 0.0898T + 79T^{2} \) |
| 83 | \( 1 + 1.67T + 83T^{2} \) |
| 89 | \( 1 + 7.08T + 89T^{2} \) |
| 97 | \( 1 + 6.48T + 97T^{2} \) |
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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
−8.286435679320853885659781610584, −7.33702822235909200301343359435, −6.70203342159280199110558466379, −5.99410555803691164158750920430, −5.35019173302441457559114509584, −4.72673716781258490103006251804, −3.31723088642282412218216783875, −2.43184726327102436928798451665, −2.11791722287105439648882061637, −0.25447927845808531973715817117,
0.25447927845808531973715817117, 2.11791722287105439648882061637, 2.43184726327102436928798451665, 3.31723088642282412218216783875, 4.72673716781258490103006251804, 5.35019173302441457559114509584, 5.99410555803691164158750920430, 6.70203342159280199110558466379, 7.33702822235909200301343359435, 8.286435679320853885659781610584