L(s) = 1 | − 3-s − 4.48·7-s + 9-s − 1.20·11-s + 13-s − 3·17-s − 7.69·19-s + 4.48·21-s − 5.69·23-s − 27-s − 7.41·29-s − 1.20·31-s + 1.20·33-s + 0.719·37-s − 39-s + 1.28·41-s − 1.28·43-s − 5.92·47-s + 13.1·49-s + 3·51-s − 6.69·53-s + 7.69·57-s + 5.92·59-s + 9.41·61-s − 4.48·63-s − 11.2·67-s + 5.69·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.69·7-s + 0.333·9-s − 0.363·11-s + 0.277·13-s − 0.727·17-s − 1.76·19-s + 0.979·21-s − 1.18·23-s − 0.192·27-s − 1.37·29-s − 0.216·31-s + 0.209·33-s + 0.118·37-s − 0.160·39-s + 0.199·41-s − 0.195·43-s − 0.864·47-s + 1.87·49-s + 0.420·51-s − 0.919·53-s + 1.01·57-s + 0.771·59-s + 1.20·61-s − 0.565·63-s − 1.36·67-s + 0.685·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2446579303\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2446579303\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + 4.48T + 7T^{2} \) |
| 11 | \( 1 + 1.20T + 11T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 + 7.69T + 19T^{2} \) |
| 23 | \( 1 + 5.69T + 23T^{2} \) |
| 29 | \( 1 + 7.41T + 29T^{2} \) |
| 31 | \( 1 + 1.20T + 31T^{2} \) |
| 37 | \( 1 - 0.719T + 37T^{2} \) |
| 41 | \( 1 - 1.28T + 41T^{2} \) |
| 43 | \( 1 + 1.28T + 43T^{2} \) |
| 47 | \( 1 + 5.92T + 47T^{2} \) |
| 53 | \( 1 + 6.69T + 53T^{2} \) |
| 59 | \( 1 - 5.92T + 59T^{2} \) |
| 61 | \( 1 - 9.41T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 - 5.13T + 71T^{2} \) |
| 73 | \( 1 + 5.13T + 73T^{2} \) |
| 79 | \( 1 - 2.71T + 79T^{2} \) |
| 83 | \( 1 + 4.89T + 83T^{2} \) |
| 89 | \( 1 + 0.412T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−7.76130401350150217600924698937, −6.96888163212877470632315285502, −6.28151000040561662183816427258, −6.08960533366683173105032132290, −5.14705108993423152160153539752, −4.13559238269110737022208862734, −3.70423931910554467302953899005, −2.65589169370230981187220115398, −1.84295891356769524160335010379, −0.23763168854140627782972561740,
0.23763168854140627782972561740, 1.84295891356769524160335010379, 2.65589169370230981187220115398, 3.70423931910554467302953899005, 4.13559238269110737022208862734, 5.14705108993423152160153539752, 6.08960533366683173105032132290, 6.28151000040561662183816427258, 6.96888163212877470632315285502, 7.76130401350150217600924698937