L(s) = 1 | + 3-s + 9-s − 13-s − 17-s + 6·19-s + 9·23-s + 27-s + 7·29-s − 7·31-s + 4·37-s − 39-s + 7·41-s − 43-s + 12·47-s − 51-s − 9·53-s + 6·57-s − 3·59-s + 9·61-s − 4·67-s + 9·69-s − 12·71-s + 8·73-s + 4·79-s + 81-s − 9·83-s + 7·87-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.277·13-s − 0.242·17-s + 1.37·19-s + 1.87·23-s + 0.192·27-s + 1.29·29-s − 1.25·31-s + 0.657·37-s − 0.160·39-s + 1.09·41-s − 0.152·43-s + 1.75·47-s − 0.140·51-s − 1.23·53-s + 0.794·57-s − 0.390·59-s + 1.15·61-s − 0.488·67-s + 1.08·69-s − 1.42·71-s + 0.936·73-s + 0.450·79-s + 1/9·81-s − 0.987·83-s + 0.750·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.423727540\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.423727540\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 9 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 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
−15.13243724544942, −14.50883849846204, −14.24015561062237, −13.57640918289926, −13.08939241112418, −12.57831035242119, −12.05816280361158, −11.31668483996953, −10.93817725961928, −10.27262636805181, −9.631980098640743, −9.139161257372018, −8.787132430068557, −7.995890217151670, −7.362637487852216, −7.122089631554167, −6.296618069863001, −5.570879150494142, −4.956894878704273, −4.392812573526136, −3.576055796775109, −2.934175858460487, −2.466300662611421, −1.407773699822123, −0.7413666780911526,
0.7413666780911526, 1.407773699822123, 2.466300662611421, 2.934175858460487, 3.576055796775109, 4.392812573526136, 4.956894878704273, 5.570879150494142, 6.296618069863001, 7.122089631554167, 7.362637487852216, 7.995890217151670, 8.787132430068557, 9.139161257372018, 9.631980098640743, 10.27262636805181, 10.93817725961928, 11.31668483996953, 12.05816280361158, 12.57831035242119, 13.08939241112418, 13.57640918289926, 14.24015561062237, 14.50883849846204, 15.13243724544942