| L(s) = 1 | + 1.84·3-s + 1.41·5-s − 0.765·7-s + 2.41·9-s − 0.765·11-s + 2.61·15-s − 1.84·19-s − 1.41·21-s + 1.00·25-s + 2.61·27-s − 1.41·33-s − 1.08·35-s − 1.41·37-s − 41-s + 3.41·45-s + 1.84·47-s − 0.414·49-s − 1.08·55-s − 3.41·57-s − 1.84·63-s − 0.765·67-s − 1.84·71-s + 1.41·73-s + 1.84·75-s + 0.585·77-s + 1.84·79-s + 2.41·81-s + ⋯ |
| L(s) = 1 | + 1.84·3-s + 1.41·5-s − 0.765·7-s + 2.41·9-s − 0.765·11-s + 2.61·15-s − 1.84·19-s − 1.41·21-s + 1.00·25-s + 2.61·27-s − 1.41·33-s − 1.08·35-s − 1.41·37-s − 41-s + 3.41·45-s + 1.84·47-s − 0.414·49-s − 1.08·55-s − 3.41·57-s − 1.84·63-s − 0.765·67-s − 1.84·71-s + 1.41·73-s + 1.84·75-s + 0.585·77-s + 1.84·79-s + 2.41·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.523806908\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.523806908\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - 1.84T + T^{2} \) |
| 5 | \( 1 - 1.41T + T^{2} \) |
| 7 | \( 1 + 0.765T + T^{2} \) |
| 11 | \( 1 + 0.765T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 1.84T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 1.41T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - 1.84T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + 0.765T + T^{2} \) |
| 71 | \( 1 + 1.84T + T^{2} \) |
| 73 | \( 1 - 1.41T + T^{2} \) |
| 79 | \( 1 - 1.84T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.934934173581028216458149869055, −8.591027326490353572569434467825, −7.66174469220706344432579086562, −6.81094519738511941325192114852, −6.15967571861287890282572768240, −5.08498083536035753835111024982, −4.03899756836353085265555069365, −3.13118067102228926691222432950, −2.36210336555706107002341535308, −1.79977135790871184252128714256,
1.79977135790871184252128714256, 2.36210336555706107002341535308, 3.13118067102228926691222432950, 4.03899756836353085265555069365, 5.08498083536035753835111024982, 6.15967571861287890282572768240, 6.81094519738511941325192114852, 7.66174469220706344432579086562, 8.591027326490353572569434467825, 8.934934173581028216458149869055
