L(s) = 1 | + 3-s − 3.95·5-s − 1.61·7-s + 9-s − 0.307·11-s + 6.01·13-s − 3.95·15-s + 1.81·17-s − 6.24·19-s − 1.61·21-s − 23-s + 10.6·25-s + 27-s − 29-s − 1.42·31-s − 0.307·33-s + 6.37·35-s − 1.75·37-s + 6.01·39-s + 2.68·41-s − 7.72·43-s − 3.95·45-s + 7.13·47-s − 4.39·49-s + 1.81·51-s + 4.45·53-s + 1.21·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.76·5-s − 0.610·7-s + 0.333·9-s − 0.0927·11-s + 1.66·13-s − 1.02·15-s + 0.439·17-s − 1.43·19-s − 0.352·21-s − 0.208·23-s + 2.12·25-s + 0.192·27-s − 0.185·29-s − 0.256·31-s − 0.0535·33-s + 1.07·35-s − 0.289·37-s + 0.962·39-s + 0.419·41-s − 1.17·43-s − 0.589·45-s + 1.04·47-s − 0.627·49-s + 0.253·51-s + 0.611·53-s + 0.163·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
good | 5 | \( 1 + 3.95T + 5T^{2} \) |
| 7 | \( 1 + 1.61T + 7T^{2} \) |
| 11 | \( 1 + 0.307T + 11T^{2} \) |
| 13 | \( 1 - 6.01T + 13T^{2} \) |
| 17 | \( 1 - 1.81T + 17T^{2} \) |
| 19 | \( 1 + 6.24T + 19T^{2} \) |
| 31 | \( 1 + 1.42T + 31T^{2} \) |
| 37 | \( 1 + 1.75T + 37T^{2} \) |
| 41 | \( 1 - 2.68T + 41T^{2} \) |
| 43 | \( 1 + 7.72T + 43T^{2} \) |
| 47 | \( 1 - 7.13T + 47T^{2} \) |
| 53 | \( 1 - 4.45T + 53T^{2} \) |
| 59 | \( 1 - 6.38T + 59T^{2} \) |
| 61 | \( 1 + 0.264T + 61T^{2} \) |
| 67 | \( 1 - 5.40T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 + 6.86T + 73T^{2} \) |
| 79 | \( 1 - 4.30T + 79T^{2} \) |
| 83 | \( 1 - 3.31T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 + 13.4T + 97T^{2} \) |
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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
−7.69302335168061054134036555047, −6.79810110286821645583675287047, −6.38404769930258447657297018394, −5.33295725527536126460964226141, −4.30907179954082714113945740905, −3.73440982352259341443014576270, −3.45328532571107462042702548328, −2.39772523206670641639209921758, −1.12312730284335470710835108970, 0,
1.12312730284335470710835108970, 2.39772523206670641639209921758, 3.45328532571107462042702548328, 3.73440982352259341443014576270, 4.30907179954082714113945740905, 5.33295725527536126460964226141, 6.38404769930258447657297018394, 6.79810110286821645583675287047, 7.69302335168061054134036555047