L(s) = 1 | − 3-s − 7-s + 9-s + 11-s + 2·13-s − 6·19-s + 21-s + 23-s − 27-s + 29-s + 2·31-s − 33-s + 7·37-s − 2·39-s − 8·41-s − 43-s − 2·47-s + 49-s + 14·53-s + 6·57-s − 10·59-s − 63-s − 3·67-s − 69-s + 9·71-s − 77-s − 79-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 1.37·19-s + 0.218·21-s + 0.208·23-s − 0.192·27-s + 0.185·29-s + 0.359·31-s − 0.174·33-s + 1.15·37-s − 0.320·39-s − 1.24·41-s − 0.152·43-s − 0.291·47-s + 1/7·49-s + 1.92·53-s + 0.794·57-s − 1.30·59-s − 0.125·63-s − 0.366·67-s − 0.120·69-s + 1.06·71-s − 0.113·77-s − 0.112·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.362961920\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.362961920\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.77977724282152080708159472828, −6.89054131867419447049138466846, −6.40786214824318910423889284583, −5.87743001296644206035422276886, −5.00165302165832436486674873396, −4.27990370053768247051750985511, −3.61936895414592254677060231704, −2.63910412087330599625429467204, −1.66485489597975843822728301132, −0.59725309347211011904007879845,
0.59725309347211011904007879845, 1.66485489597975843822728301132, 2.63910412087330599625429467204, 3.61936895414592254677060231704, 4.27990370053768247051750985511, 5.00165302165832436486674873396, 5.87743001296644206035422276886, 6.40786214824318910423889284583, 6.89054131867419447049138466846, 7.77977724282152080708159472828