L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s − 2·9-s − 3·11-s + 12-s − 4·13-s + 16-s − 3·17-s − 2·18-s − 5·19-s − 3·22-s − 6·23-s + 24-s − 4·26-s − 5·27-s − 2·31-s + 32-s − 3·33-s − 3·34-s − 2·36-s − 2·37-s − 5·38-s − 4·39-s + 3·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s − 2/3·9-s − 0.904·11-s + 0.288·12-s − 1.10·13-s + 1/4·16-s − 0.727·17-s − 0.471·18-s − 1.14·19-s − 0.639·22-s − 1.25·23-s + 0.204·24-s − 0.784·26-s − 0.962·27-s − 0.359·31-s + 0.176·32-s − 0.522·33-s − 0.514·34-s − 1/3·36-s − 0.328·37-s − 0.811·38-s − 0.640·39-s + 0.468·41-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.409810000821400938912682522137, −7.82315988608083832795544283712, −7.03565158477587046360278529923, −6.08705351503109341254138790112, −5.38603857261488641210316347351, −4.49973313906049110134055340472, −3.70222391402935277847702964230, −2.50196990522414289793307495614, −2.23496341365700158267997210371, 0,
2.23496341365700158267997210371, 2.50196990522414289793307495614, 3.70222391402935277847702964230, 4.49973313906049110134055340472, 5.38603857261488641210316347351, 6.08705351503109341254138790112, 7.03565158477587046360278529923, 7.82315988608083832795544283712, 8.409810000821400938912682522137