L(s) = 1 | + (0.0395 − 0.0865i)3-s + (1.02 − 1.18i)5-s + (0.648 + 0.748i)9-s + (−0.142 + 0.989i)11-s + (−0.0621 − 0.136i)15-s + (−0.786 − 0.618i)23-s + (−0.209 − 1.45i)25-s + (0.181 − 0.0533i)27-s + (−0.271 − 0.595i)31-s + (0.0800 + 0.0514i)33-s + (−0.308 − 0.356i)37-s + 1.55·45-s − 1.30·47-s + (0.415 − 0.909i)49-s + (−1.61 + 1.03i)53-s + ⋯ |
L(s) = 1 | + (0.0395 − 0.0865i)3-s + (1.02 − 1.18i)5-s + (0.648 + 0.748i)9-s + (−0.142 + 0.989i)11-s + (−0.0621 − 0.136i)15-s + (−0.786 − 0.618i)23-s + (−0.209 − 1.45i)25-s + (0.181 − 0.0533i)27-s + (−0.271 − 0.595i)31-s + (0.0800 + 0.0514i)33-s + (−0.308 − 0.356i)37-s + 1.55·45-s − 1.30·47-s + (0.415 − 0.909i)49-s + (−1.61 + 1.03i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.244130132\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.244130132\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 + (0.142 - 0.989i)T \) |
| 23 | \( 1 + (0.786 + 0.618i)T \) |
good | 3 | \( 1 + (-0.0395 + 0.0865i)T + (-0.654 - 0.755i)T^{2} \) |
| 5 | \( 1 + (-1.02 + 1.18i)T + (-0.142 - 0.989i)T^{2} \) |
| 7 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 13 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 17 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 19 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 29 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 31 | \( 1 + (0.271 + 0.595i)T + (-0.654 + 0.755i)T^{2} \) |
| 37 | \( 1 + (0.308 + 0.356i)T + (-0.142 + 0.989i)T^{2} \) |
| 41 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 43 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 47 | \( 1 + 1.30T + T^{2} \) |
| 53 | \( 1 + (1.61 - 1.03i)T + (0.415 - 0.909i)T^{2} \) |
| 59 | \( 1 + (-1.21 - 0.782i)T + (0.415 + 0.909i)T^{2} \) |
| 61 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 67 | \( 1 + (-0.252 - 1.75i)T + (-0.959 + 0.281i)T^{2} \) |
| 71 | \( 1 + (0.264 + 1.83i)T + (-0.959 + 0.281i)T^{2} \) |
| 73 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 79 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 83 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 89 | \( 1 + (0.738 - 1.61i)T + (-0.654 - 0.755i)T^{2} \) |
| 97 | \( 1 + (-0.428 + 0.494i)T + (-0.142 - 0.989i)T^{2} \) |
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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
−9.954042647139543944238467811270, −9.425163999095514131142688888060, −8.482866965185489347417032650232, −7.70134231356421130219300221180, −6.71316367292014730002898708508, −5.66040370042887656587442794899, −4.89930911159952235939783786083, −4.18533566456332004062009129437, −2.33354316913659955937506022092, −1.54837841708496927171132103802,
1.68900360270520557452273090533, 2.97957961096956897971060052517, 3.72025200709799855522063299147, 5.18196656051759068447656997494, 6.18453168939110037622514999858, 6.61024557115229771251440369293, 7.60330428610133391336156673239, 8.666543542864839743999107647519, 9.670737357570697547751946417764, 10.06084890035109153060006252313