L(s) = 1 | + 2.41·2-s − 1.41·3-s + 3.82·4-s − 3.82·5-s − 3.41·6-s + 4.41·8-s − 0.999·9-s − 9.24·10-s − 3.41·11-s − 5.41·12-s + 5.41·15-s + 2.99·16-s − 0.171·17-s − 2.41·18-s − 6·19-s − 14.6·20-s − 8.24·22-s + 1.41·23-s − 6.24·24-s + 9.65·25-s + 5.65·27-s + 9.82·29-s + 13.0·30-s − 5.41·31-s − 1.58·32-s + 4.82·33-s − 0.414·34-s + ⋯ |
L(s) = 1 | + 1.70·2-s − 0.816·3-s + 1.91·4-s − 1.71·5-s − 1.39·6-s + 1.56·8-s − 0.333·9-s − 2.92·10-s − 1.02·11-s − 1.56·12-s + 1.39·15-s + 0.749·16-s − 0.0416·17-s − 0.569·18-s − 1.37·19-s − 3.27·20-s − 1.75·22-s + 0.294·23-s − 1.27·24-s + 1.93·25-s + 1.08·27-s + 1.82·29-s + 2.38·30-s − 0.972·31-s − 0.280·32-s + 0.840·33-s − 0.0710·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.490358478\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.490358478\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 + 3.82T + 5T^{2} \) |
| 11 | \( 1 + 3.41T + 11T^{2} \) |
| 17 | \( 1 + 0.171T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 - 9.82T + 29T^{2} \) |
| 31 | \( 1 + 5.41T + 31T^{2} \) |
| 37 | \( 1 + 7.48T + 37T^{2} \) |
| 41 | \( 1 + 5.82T + 41T^{2} \) |
| 43 | \( 1 - 0.585T + 43T^{2} \) |
| 47 | \( 1 + 7.65T + 47T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 - 1.75T + 59T^{2} \) |
| 61 | \( 1 - 9.82T + 61T^{2} \) |
| 67 | \( 1 - 4.24T + 67T^{2} \) |
| 71 | \( 1 - 0.343T + 71T^{2} \) |
| 73 | \( 1 + 0.656T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 - 7.31T + 89T^{2} \) |
| 97 | \( 1 - 5.17T + 97T^{2} \) |
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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
−7.58983832545271253799539452716, −6.66289786586691553307961267356, −6.51109295440330560768832552836, −5.38367074680879136085848472525, −4.96538456529653155805788404122, −4.45135487511530341072930696162, −3.60095906883833278867613177371, −3.10088772763732643472640546250, −2.17254594346472368837632573416, −0.44926001730340244032505801707,
0.44926001730340244032505801707, 2.17254594346472368837632573416, 3.10088772763732643472640546250, 3.60095906883833278867613177371, 4.45135487511530341072930696162, 4.96538456529653155805788404122, 5.38367074680879136085848472525, 6.51109295440330560768832552836, 6.66289786586691553307961267356, 7.58983832545271253799539452716