L(s) = 1 | − 3-s + 2·5-s + 4·7-s + 9-s − 4·11-s − 2·13-s − 2·15-s − 6·17-s + 4·19-s − 4·21-s − 25-s − 27-s + 2·29-s − 4·31-s + 4·33-s + 8·35-s − 2·37-s + 2·39-s + 2·41-s − 4·43-s + 2·45-s − 8·47-s + 9·49-s + 6·51-s + 10·53-s − 8·55-s − 4·57-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1.51·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.516·15-s − 1.45·17-s + 0.917·19-s − 0.872·21-s − 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.718·31-s + 0.696·33-s + 1.35·35-s − 0.328·37-s + 0.320·39-s + 0.312·41-s − 0.609·43-s + 0.298·45-s − 1.16·47-s + 9/7·49-s + 0.840·51-s + 1.37·53-s − 1.07·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.001077380\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.001077380\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.87948688478378416848706681259, −13.02862772705358459630904610270, −11.67525385332011727244436354557, −10.84849442743057013529606410604, −9.826637660192491869357018654726, −8.391046533691483698004382219288, −7.17212008872630953090710242182, −5.56983356275681900785379754785, −4.77602747706575131165960838209, −2.09803110754033007053716721218,
2.09803110754033007053716721218, 4.77602747706575131165960838209, 5.56983356275681900785379754785, 7.17212008872630953090710242182, 8.391046533691483698004382219288, 9.826637660192491869357018654726, 10.84849442743057013529606410604, 11.67525385332011727244436354557, 13.02862772705358459630904610270, 13.87948688478378416848706681259