L(s) = 1 | + (−1.78 − 0.523i)3-s + (0.975 − 0.627i)5-s + (2.05 + 1.32i)9-s + (0.415 + 0.909i)11-s + (−2.06 + 0.606i)15-s + (0.580 − 0.814i)23-s + (0.143 − 0.314i)25-s + (−1.75 − 2.03i)27-s + (1.70 − 0.500i)31-s + (−0.264 − 1.83i)33-s + (−0.550 − 0.353i)37-s + 2.83·45-s + 1.68·47-s + (−0.959 − 0.281i)49-s + (0.186 − 1.29i)53-s + ⋯ |
L(s) = 1 | + (−1.78 − 0.523i)3-s + (0.975 − 0.627i)5-s + (2.05 + 1.32i)9-s + (0.415 + 0.909i)11-s + (−2.06 + 0.606i)15-s + (0.580 − 0.814i)23-s + (0.143 − 0.314i)25-s + (−1.75 − 2.03i)27-s + (1.70 − 0.500i)31-s + (−0.264 − 1.83i)33-s + (−0.550 − 0.353i)37-s + 2.83·45-s + 1.68·47-s + (−0.959 − 0.281i)49-s + (0.186 − 1.29i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.740 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.740 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7235918170\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7235918170\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 + (-0.415 - 0.909i)T \) |
| 23 | \( 1 + (-0.580 + 0.814i)T \) |
good | 3 | \( 1 + (1.78 + 0.523i)T + (0.841 + 0.540i)T^{2} \) |
| 5 | \( 1 + (-0.975 + 0.627i)T + (0.415 - 0.909i)T^{2} \) |
| 7 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 13 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 17 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 19 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 29 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 31 | \( 1 + (-1.70 + 0.500i)T + (0.841 - 0.540i)T^{2} \) |
| 37 | \( 1 + (0.550 + 0.353i)T + (0.415 + 0.909i)T^{2} \) |
| 41 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 43 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 47 | \( 1 - 1.68T + T^{2} \) |
| 53 | \( 1 + (-0.186 + 1.29i)T + (-0.959 - 0.281i)T^{2} \) |
| 59 | \( 1 + (0.279 + 1.94i)T + (-0.959 + 0.281i)T^{2} \) |
| 61 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 67 | \( 1 + (0.653 - 1.43i)T + (-0.654 - 0.755i)T^{2} \) |
| 71 | \( 1 + (0.827 - 1.81i)T + (-0.654 - 0.755i)T^{2} \) |
| 73 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 79 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 83 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 89 | \( 1 + (-1.50 - 0.442i)T + (0.841 + 0.540i)T^{2} \) |
| 97 | \( 1 + (1.49 - 0.961i)T + (0.415 - 0.909i)T^{2} \) |
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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
−10.10003951094501347049259503176, −9.565553246113512161219827611680, −8.376682684691236663599242939515, −7.15384782205572984097468035250, −6.56948807860227876452804712691, −5.77324584614099675641224288504, −5.04525955617853704353805791920, −4.34110563688094718670185024805, −2.15661043249775789286381998682, −1.09158849883351725535540448693,
1.26148405251957857343287873247, 3.06049570239984864339456134687, 4.33185455203400020049958875029, 5.27411461096678353176741274900, 6.08666719589467035632042644317, 6.40477696759621647563543152999, 7.42383167740252787014917971704, 8.936981514896942291508407334038, 9.694330985515113643573226046886, 10.59136349681379346336062596791