L(s) = 1 | − 1.48·3-s − 5-s − 3.53·7-s − 0.788·9-s − 0.193·11-s − 1.78·13-s + 1.48·15-s + 0.112·17-s − 1.44·19-s + 5.25·21-s + 7.37·23-s + 25-s + 5.63·27-s − 1.99·29-s − 2.75·31-s + 0.287·33-s + 3.53·35-s − 5.38·37-s + 2.65·39-s + 9.08·41-s + 7.95·43-s + 0.788·45-s − 9.63·47-s + 5.49·49-s − 0.167·51-s + 1.21·53-s + 0.193·55-s + ⋯ |
L(s) = 1 | − 0.858·3-s − 0.447·5-s − 1.33·7-s − 0.262·9-s − 0.0583·11-s − 0.495·13-s + 0.383·15-s + 0.0273·17-s − 0.331·19-s + 1.14·21-s + 1.53·23-s + 0.200·25-s + 1.08·27-s − 0.370·29-s − 0.495·31-s + 0.0500·33-s + 0.597·35-s − 0.885·37-s + 0.425·39-s + 1.41·41-s + 1.21·43-s + 0.117·45-s − 1.40·47-s + 0.784·49-s − 0.0234·51-s + 0.167·53-s + 0.0260·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 + 1.48T + 3T^{2} \) |
| 7 | \( 1 + 3.53T + 7T^{2} \) |
| 11 | \( 1 + 0.193T + 11T^{2} \) |
| 13 | \( 1 + 1.78T + 13T^{2} \) |
| 17 | \( 1 - 0.112T + 17T^{2} \) |
| 19 | \( 1 + 1.44T + 19T^{2} \) |
| 23 | \( 1 - 7.37T + 23T^{2} \) |
| 29 | \( 1 + 1.99T + 29T^{2} \) |
| 31 | \( 1 + 2.75T + 31T^{2} \) |
| 37 | \( 1 + 5.38T + 37T^{2} \) |
| 41 | \( 1 - 9.08T + 41T^{2} \) |
| 43 | \( 1 - 7.95T + 43T^{2} \) |
| 47 | \( 1 + 9.63T + 47T^{2} \) |
| 53 | \( 1 - 1.21T + 53T^{2} \) |
| 59 | \( 1 + 4.20T + 59T^{2} \) |
| 61 | \( 1 - 6.14T + 61T^{2} \) |
| 67 | \( 1 - 8.98T + 67T^{2} \) |
| 71 | \( 1 + 7.09T + 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 - 5.03T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 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
−7.30388268792238752058923385570, −6.69341824459596485393208929047, −6.20333033288119274969627153191, −5.38424161883485390880190712808, −4.85633116557871954558573243522, −3.83816445295995713895867868344, −3.16206384680738968066625255628, −2.38717574778847880180105262598, −0.857228154443025368527627935911, 0,
0.857228154443025368527627935911, 2.38717574778847880180105262598, 3.16206384680738968066625255628, 3.83816445295995713895867868344, 4.85633116557871954558573243522, 5.38424161883485390880190712808, 6.20333033288119274969627153191, 6.69341824459596485393208929047, 7.30388268792238752058923385570