| L(s) = 1 | − 2·4-s − 5-s + 3·7-s − 3·9-s − 5·11-s + 4·16-s − 7·17-s + 2·20-s − 4·23-s − 4·25-s − 6·28-s − 3·35-s + 6·36-s − 43-s + 10·44-s + 3·45-s + 13·47-s + 2·49-s + 5·55-s + 15·61-s − 9·63-s − 8·64-s + 14·68-s − 11·73-s − 15·77-s − 4·80-s + 9·81-s + ⋯ |
| L(s) = 1 | − 4-s − 0.447·5-s + 1.13·7-s − 9-s − 1.50·11-s + 16-s − 1.69·17-s + 0.447·20-s − 0.834·23-s − 4/5·25-s − 1.13·28-s − 0.507·35-s + 36-s − 0.152·43-s + 1.50·44-s + 0.447·45-s + 1.89·47-s + 2/7·49-s + 0.674·55-s + 1.92·61-s − 1.13·63-s − 64-s + 1.69·68-s − 1.28·73-s − 1.70·77-s − 0.447·80-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 19 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 15 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−11.04192157513365594505715826533, −10.14137137498683452879063403529, −8.818740829912344599292769500918, −8.301202088179616535243949193699, −7.53951354613441402441563530018, −5.80534064056569741924876016031, −4.96494399417687847585027962398, −4.04609256039116080936481593666, −2.40706237497488198685499903779, 0,
2.40706237497488198685499903779, 4.04609256039116080936481593666, 4.96494399417687847585027962398, 5.80534064056569741924876016031, 7.53951354613441402441563530018, 8.301202088179616535243949193699, 8.818740829912344599292769500918, 10.14137137498683452879063403529, 11.04192157513365594505715826533
