L(s) = 1 | + (0.809 + 0.587i)5-s + (−0.309 + 0.951i)9-s + (−1.11 − 0.363i)13-s + (1.11 − 1.53i)17-s + (0.309 + 0.951i)25-s + (−0.5 + 0.363i)29-s + (−1.80 − 0.587i)37-s + (0.5 − 1.53i)41-s + (−0.809 + 0.587i)45-s − 49-s + (−0.690 − 0.951i)53-s + (0.5 + 1.53i)61-s + (−0.690 − 0.951i)65-s + (−1.11 + 0.363i)73-s + (−0.809 − 0.587i)81-s + ⋯ |
L(s) = 1 | + (0.809 + 0.587i)5-s + (−0.309 + 0.951i)9-s + (−1.11 − 0.363i)13-s + (1.11 − 1.53i)17-s + (0.309 + 0.951i)25-s + (−0.5 + 0.363i)29-s + (−1.80 − 0.587i)37-s + (0.5 − 1.53i)41-s + (−0.809 + 0.587i)45-s − 49-s + (−0.690 − 0.951i)53-s + (0.5 + 1.53i)61-s + (−0.690 − 0.951i)65-s + (−1.11 + 0.363i)73-s + (−0.809 − 0.587i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8987725254\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8987725254\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.809 - 0.587i)T \) |
good | 3 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (1.11 + 0.363i)T + (0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (-1.11 + 1.53i)T + (-0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (1.80 + 0.587i)T + (0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (0.690 + 0.951i)T + (-0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 71 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (1.11 - 0.363i)T + (0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (-1.11 - 1.53i)T + (-0.309 + 0.951i)T^{2} \) |
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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
−11.47447693944911404871018336081, −10.49291110913306791471921002321, −9.895876285656303192517067452653, −8.952153147393279456436444436742, −7.60550967556955935521421202044, −7.05623045946663616135209441725, −5.56969426992634278410032912524, −5.06067267484679542299630247484, −3.18756660000288309152276272385, −2.16399591947852327969843929509,
1.67747576978274899193502685885, 3.28986863138439458539880391728, 4.65983919067586011978452406047, 5.75806364247508604987487712696, 6.52427110502955273373945286963, 7.84414524235281552958243320580, 8.816837374992362635113816311346, 9.676549757626723142209792648681, 10.26448278078356289299910946481, 11.60109897579176044144238305398