| L(s) = 1 | + 2.39·3-s + 1.16·5-s − 7-s + 2.73·9-s − 3.38·11-s + 2.57·13-s + 2.78·15-s − 17-s − 6.30·19-s − 2.39·21-s − 8.41·23-s − 3.65·25-s − 0.623·27-s + 5.38·29-s + 2.05·31-s − 8.11·33-s − 1.16·35-s − 3.25·37-s + 6.17·39-s + 7.90·41-s − 8.38·43-s + 3.18·45-s + 4.48·47-s + 49-s − 2.39·51-s + 1.33·53-s − 3.93·55-s + ⋯ |
| L(s) = 1 | + 1.38·3-s + 0.519·5-s − 0.377·7-s + 0.913·9-s − 1.02·11-s + 0.715·13-s + 0.718·15-s − 0.242·17-s − 1.44·19-s − 0.522·21-s − 1.75·23-s − 0.730·25-s − 0.120·27-s + 1.00·29-s + 0.368·31-s − 1.41·33-s − 0.196·35-s − 0.535·37-s + 0.989·39-s + 1.23·41-s − 1.27·43-s + 0.474·45-s + 0.653·47-s + 0.142·49-s − 0.335·51-s + 0.183·53-s − 0.530·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 2.39T + 3T^{2} \) |
| 5 | \( 1 - 1.16T + 5T^{2} \) |
| 11 | \( 1 + 3.38T + 11T^{2} \) |
| 13 | \( 1 - 2.57T + 13T^{2} \) |
| 19 | \( 1 + 6.30T + 19T^{2} \) |
| 23 | \( 1 + 8.41T + 23T^{2} \) |
| 29 | \( 1 - 5.38T + 29T^{2} \) |
| 31 | \( 1 - 2.05T + 31T^{2} \) |
| 37 | \( 1 + 3.25T + 37T^{2} \) |
| 41 | \( 1 - 7.90T + 41T^{2} \) |
| 43 | \( 1 + 8.38T + 43T^{2} \) |
| 47 | \( 1 - 4.48T + 47T^{2} \) |
| 53 | \( 1 - 1.33T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 - 3.78T + 61T^{2} \) |
| 67 | \( 1 - 3.57T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 - 1.13T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 + 17.5T + 83T^{2} \) |
| 89 | \( 1 + 7.74T + 89T^{2} \) |
| 97 | \( 1 + 8.44T + 97T^{2} \) |
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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
−7.933006739509674285291513294860, −6.85069589330034550313790205649, −6.16013752750514647548659274391, −5.58067027604407062577912166137, −4.39517196128295950093334351060, −3.88682167798054317988311265563, −2.93882944246452114614097312852, −2.35530048502618645618380651509, −1.68255661731242692862343202181, 0,
1.68255661731242692862343202181, 2.35530048502618645618380651509, 2.93882944246452114614097312852, 3.88682167798054317988311265563, 4.39517196128295950093334351060, 5.58067027604407062577912166137, 6.16013752750514647548659274391, 6.85069589330034550313790205649, 7.933006739509674285291513294860
