L(s) = 1 | + (0.707 − 0.707i)3-s + (1.50 + 1.65i)5-s + 2.58·7-s − 1.00i·9-s + (4.39 − 4.39i)11-s + (0.417 − 0.417i)13-s + (2.23 + 0.110i)15-s − 4.40i·17-s + (−4.53 − 4.53i)19-s + (1.83 − 1.83i)21-s + 0.281·23-s + (−0.494 + 4.97i)25-s + (−0.707 − 0.707i)27-s + (−3.73 − 3.73i)29-s + 3.05·31-s + ⋯ |
L(s) = 1 | + (0.408 − 0.408i)3-s + (0.671 + 0.741i)5-s + 0.978·7-s − 0.333i·9-s + (1.32 − 1.32i)11-s + (0.115 − 0.115i)13-s + (0.576 + 0.0286i)15-s − 1.06i·17-s + (−1.04 − 1.04i)19-s + (0.399 − 0.399i)21-s + 0.0586·23-s + (−0.0989 + 0.995i)25-s + (−0.136 − 0.136i)27-s + (−0.693 − 0.693i)29-s + 0.548·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.717 + 0.697i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.717 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.695697427\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.695697427\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (-1.50 - 1.65i)T \) |
good | 7 | \( 1 - 2.58T + 7T^{2} \) |
| 11 | \( 1 + (-4.39 + 4.39i)T - 11iT^{2} \) |
| 13 | \( 1 + (-0.417 + 0.417i)T - 13iT^{2} \) |
| 17 | \( 1 + 4.40iT - 17T^{2} \) |
| 19 | \( 1 + (4.53 + 4.53i)T + 19iT^{2} \) |
| 23 | \( 1 - 0.281T + 23T^{2} \) |
| 29 | \( 1 + (3.73 + 3.73i)T + 29iT^{2} \) |
| 31 | \( 1 - 3.05T + 31T^{2} \) |
| 37 | \( 1 + (5.26 + 5.26i)T + 37iT^{2} \) |
| 41 | \( 1 - 5.16iT - 41T^{2} \) |
| 43 | \( 1 + (-2.66 - 2.66i)T + 43iT^{2} \) |
| 47 | \( 1 - 7.45iT - 47T^{2} \) |
| 53 | \( 1 + (2.89 + 2.89i)T + 53iT^{2} \) |
| 59 | \( 1 + (4.60 - 4.60i)T - 59iT^{2} \) |
| 61 | \( 1 + (-0.211 - 0.211i)T + 61iT^{2} \) |
| 67 | \( 1 + (-7.17 + 7.17i)T - 67iT^{2} \) |
| 71 | \( 1 - 15.9iT - 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 + 4.53T + 79T^{2} \) |
| 83 | \( 1 + (-4.56 + 4.56i)T - 83iT^{2} \) |
| 89 | \( 1 - 10.2iT - 89T^{2} \) |
| 97 | \( 1 - 6.78iT - 97T^{2} \) |
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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
−9.090438397725272326129119919800, −8.376024053967059980784523235321, −7.53864746053579681614517020548, −6.64320350150276183898587704952, −6.15960893621452698567353237276, −5.13167620887939193464670767466, −4.03395702324562646170240193664, −3.02054990340639642981646723742, −2.14240569755921785482907768477, −1.00002023228593297529059685215,
1.69814370919847249447273332961, 1.85020120621109428575174594788, 3.72321003435446718375252381774, 4.38012677601576764905365223462, 5.08027845495239601849316075759, 6.07654761123664032232946278346, 6.90631627513216016084371389544, 8.004934277803885254955962738726, 8.613545502776969545031916449448, 9.193122432846053511017564598009