L(s) = 1 | − 3-s + 4-s + 9-s − 12-s + 16-s + 25-s − 27-s + 36-s − 2·43-s − 48-s − 49-s − 2·61-s + 64-s − 75-s − 2·79-s + 81-s + 100-s − 2·103-s − 108-s + ⋯ |
L(s) = 1 | − 3-s + 4-s + 9-s − 12-s + 16-s + 25-s − 27-s + 36-s − 2·43-s − 48-s − 49-s − 2·61-s + 64-s − 75-s − 2·79-s + 81-s + 100-s − 2·103-s − 108-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8317882095\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8317882095\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 + T )^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 + T )^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( ( 1 + T )^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( 1 + T^{2} \) |
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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
−11.13367580834881952500247662593, −10.49163999321511295089647941292, −9.651633738726069267537770192589, −8.282933462069901628483253293923, −7.20996448133866159023524376902, −6.55691982982863666494491758146, −5.67248665125553439266503010963, −4.63712375411729631409827007592, −3.16476648458228068363240929490, −1.59349101491784544382124003930,
1.59349101491784544382124003930, 3.16476648458228068363240929490, 4.63712375411729631409827007592, 5.67248665125553439266503010963, 6.55691982982863666494491758146, 7.20996448133866159023524376902, 8.282933462069901628483253293923, 9.651633738726069267537770192589, 10.49163999321511295089647941292, 11.13367580834881952500247662593