L(s) = 1 | + 1.09·3-s − 0.0446·5-s + 4.14·7-s − 1.79·9-s − 6.33·11-s + 2.84·13-s − 0.0490·15-s + 3.68·17-s − 2.76·19-s + 4.55·21-s − 6.82·23-s − 4.99·25-s − 5.26·27-s + 9.76·29-s − 9.04·31-s − 6.95·33-s − 0.185·35-s − 7.68·37-s + 3.11·39-s + 4.70·41-s − 0.928·43-s + 0.0802·45-s − 1.75·47-s + 10.1·49-s + 4.04·51-s − 6.88·53-s + 0.283·55-s + ⋯ |
L(s) = 1 | + 0.633·3-s − 0.0199·5-s + 1.56·7-s − 0.598·9-s − 1.91·11-s + 0.788·13-s − 0.0126·15-s + 0.893·17-s − 0.633·19-s + 0.993·21-s − 1.42·23-s − 0.999·25-s − 1.01·27-s + 1.81·29-s − 1.62·31-s − 1.21·33-s − 0.0313·35-s − 1.26·37-s + 0.499·39-s + 0.734·41-s − 0.141·43-s + 0.0119·45-s − 0.256·47-s + 1.45·49-s + 0.566·51-s − 0.945·53-s + 0.0381·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 751 | \( 1 + T \) |
good | 3 | \( 1 - 1.09T + 3T^{2} \) |
| 5 | \( 1 + 0.0446T + 5T^{2} \) |
| 7 | \( 1 - 4.14T + 7T^{2} \) |
| 11 | \( 1 + 6.33T + 11T^{2} \) |
| 13 | \( 1 - 2.84T + 13T^{2} \) |
| 17 | \( 1 - 3.68T + 17T^{2} \) |
| 19 | \( 1 + 2.76T + 19T^{2} \) |
| 23 | \( 1 + 6.82T + 23T^{2} \) |
| 29 | \( 1 - 9.76T + 29T^{2} \) |
| 31 | \( 1 + 9.04T + 31T^{2} \) |
| 37 | \( 1 + 7.68T + 37T^{2} \) |
| 41 | \( 1 - 4.70T + 41T^{2} \) |
| 43 | \( 1 + 0.928T + 43T^{2} \) |
| 47 | \( 1 + 1.75T + 47T^{2} \) |
| 53 | \( 1 + 6.88T + 53T^{2} \) |
| 59 | \( 1 - 5.45T + 59T^{2} \) |
| 61 | \( 1 + 1.13T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 + 9.49T + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 - 6.76T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 - 4.86T + 89T^{2} \) |
| 97 | \( 1 + 7.30T + 97T^{2} \) |
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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
−7.85273951114659133807556757135, −7.46646086257452510833935691285, −6.07091699187193551116309791043, −5.55028604760104249937734707405, −4.89732337136900045084811460347, −4.01984178533684971249298201445, −3.13328612272573268238164367153, −2.27885183158081141133528216388, −1.60608627064810855260565381085, 0,
1.60608627064810855260565381085, 2.27885183158081141133528216388, 3.13328612272573268238164367153, 4.01984178533684971249298201445, 4.89732337136900045084811460347, 5.55028604760104249937734707405, 6.07091699187193551116309791043, 7.46646086257452510833935691285, 7.85273951114659133807556757135