| L(s) = 1 | − 7-s + 11-s + 2·13-s − 2·17-s + 2·29-s + 4·31-s − 6·37-s − 6·41-s + 12·43-s − 4·47-s + 49-s + 6·53-s − 4·59-s − 10·61-s − 4·67-s + 8·71-s + 2·73-s − 77-s + 16·83-s + 6·89-s − 2·91-s + 6·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.301·11-s + 0.554·13-s − 0.485·17-s + 0.371·29-s + 0.718·31-s − 0.986·37-s − 0.937·41-s + 1.82·43-s − 0.583·47-s + 1/7·49-s + 0.824·53-s − 0.520·59-s − 1.28·61-s − 0.488·67-s + 0.949·71-s + 0.234·73-s − 0.113·77-s + 1.75·83-s + 0.635·89-s − 0.209·91-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 277200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.02885943165011, −12.47222012418165, −12.04658578208310, −11.73770477780256, −11.07889172689346, −10.65177313630761, −10.34435140642514, −9.746237340667151, −9.175241056696093, −8.929699808716377, −8.402472107971936, −7.837227686678387, −7.415810833062885, −6.754434899265233, −6.368272890624633, −6.081476746276465, −5.281244644573821, −4.940283691792911, −4.227553154379197, −3.830363719521475, −3.259690355973580, −2.696474056641958, −2.098008062473807, −1.415043101500382, −0.7958229996387536, 0,
0.7958229996387536, 1.415043101500382, 2.098008062473807, 2.696474056641958, 3.259690355973580, 3.830363719521475, 4.227553154379197, 4.940283691792911, 5.281244644573821, 6.081476746276465, 6.368272890624633, 6.754434899265233, 7.415810833062885, 7.837227686678387, 8.402472107971936, 8.929699808716377, 9.175241056696093, 9.746237340667151, 10.34435140642514, 10.65177313630761, 11.07889172689346, 11.73770477780256, 12.04658578208310, 12.47222012418165, 13.02885943165011
