| L(s) = 1 | − 3·3-s − 2.13·7-s + 9·9-s + 24.4·11-s − 34.0·13-s + 123.·17-s + 9.17·19-s + 6.39·21-s + 142.·23-s − 27·27-s + 140.·29-s − 158.·31-s − 73.3·33-s − 58.5·37-s + 102.·39-s − 108.·41-s − 246.·43-s + 466.·47-s − 338.·49-s − 369.·51-s + 312.·53-s − 27.5·57-s − 410.·59-s − 44.5·61-s − 19.1·63-s − 368.·67-s − 426.·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.115·7-s + 0.333·9-s + 0.670·11-s − 0.726·13-s + 1.75·17-s + 0.110·19-s + 0.0664·21-s + 1.28·23-s − 0.192·27-s + 0.899·29-s − 0.919·31-s − 0.386·33-s − 0.260·37-s + 0.419·39-s − 0.414·41-s − 0.874·43-s + 1.44·47-s − 0.986·49-s − 1.01·51-s + 0.810·53-s − 0.0639·57-s − 0.906·59-s − 0.0935·61-s − 0.0383·63-s − 0.671·67-s − 0.744·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.892099651\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.892099651\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2.13T + 343T^{2} \) |
| 11 | \( 1 - 24.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 34.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 123.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 9.17T + 6.85e3T^{2} \) |
| 23 | \( 1 - 142.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 140.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 158.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 58.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 108.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 246.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 466.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 312.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 410.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 44.5T + 2.26e5T^{2} \) |
| 67 | \( 1 + 368.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 108.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 627.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 196.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 107.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 685.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 73.7T + 9.12e5T^{2} \) |
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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
−8.681774877945550579162497077069, −7.66152532038120347026760459962, −7.08822715532891634201450032133, −6.28685541890895463278820098716, −5.40391903127241154127524502919, −4.83496327855757623202050540668, −3.72648294647191813166796082234, −2.91719526339620060340391482866, −1.56157093043325149807622077268, −0.66677101232613254400132192256,
0.66677101232613254400132192256, 1.56157093043325149807622077268, 2.91719526339620060340391482866, 3.72648294647191813166796082234, 4.83496327855757623202050540668, 5.40391903127241154127524502919, 6.28685541890895463278820098716, 7.08822715532891634201450032133, 7.66152532038120347026760459962, 8.681774877945550579162497077069
