L(s) = 1 | + 2-s + 1.41·3-s + 4-s + 1.41·6-s + 8-s − 0.999·9-s + 1.41·12-s + 4.24·13-s + 16-s − 5.65·17-s − 0.999·18-s + 4.24·19-s + 6·23-s + 1.41·24-s + 4.24·26-s − 5.65·27-s + 6·29-s + 8.48·31-s + 32-s − 5.65·34-s − 0.999·36-s + 6·37-s + 4.24·38-s + 6·39-s − 8.48·41-s + 12·43-s + 6·46-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.816·3-s + 0.5·4-s + 0.577·6-s + 0.353·8-s − 0.333·9-s + 0.408·12-s + 1.17·13-s + 0.250·16-s − 1.37·17-s − 0.235·18-s + 0.973·19-s + 1.25·23-s + 0.288·24-s + 0.832·26-s − 1.08·27-s + 1.11·29-s + 1.52·31-s + 0.176·32-s − 0.970·34-s − 0.166·36-s + 0.986·37-s + 0.688·38-s + 0.960·39-s − 1.32·41-s + 1.82·43-s + 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.924420499\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.924420499\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 + 5.65T + 17T^{2} \) |
| 19 | \( 1 - 4.24T + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 8.48T + 41T^{2} \) |
| 43 | \( 1 - 12T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 4.24T + 59T^{2} \) |
| 61 | \( 1 - 4.24T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 8.48T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 15.5T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−8.766798130805968109046704672480, −8.312898819076376913363095759417, −7.37271998442593291735691271897, −6.53312344544757060796474865140, −5.86768002058265817649048487513, −4.83937694972565554770101594754, −4.07497151286776616042894507502, −3.07642434613909907597530519817, −2.57143776317776818554336913010, −1.18784989357059655712253803189,
1.18784989357059655712253803189, 2.57143776317776818554336913010, 3.07642434613909907597530519817, 4.07497151286776616042894507502, 4.83937694972565554770101594754, 5.86768002058265817649048487513, 6.53312344544757060796474865140, 7.37271998442593291735691271897, 8.312898819076376913363095759417, 8.766798130805968109046704672480