L(s) = 1 | + 3-s + 7-s + 9-s − 11-s + 2·13-s + 6·19-s + 21-s − 23-s + 27-s + 29-s − 2·31-s − 33-s + 7·37-s + 2·39-s − 8·41-s + 43-s + 2·47-s + 49-s + 14·53-s + 6·57-s + 10·59-s + 63-s + 3·67-s − 69-s − 9·71-s − 77-s + 79-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 1.37·19-s + 0.218·21-s − 0.208·23-s + 0.192·27-s + 0.185·29-s − 0.359·31-s − 0.174·33-s + 1.15·37-s + 0.320·39-s − 1.24·41-s + 0.152·43-s + 0.291·47-s + 1/7·49-s + 1.92·53-s + 0.794·57-s + 1.30·59-s + 0.125·63-s + 0.366·67-s − 0.120·69-s − 1.06·71-s − 0.113·77-s + 0.112·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.422482730\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.422482730\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.997461625279297363034935811227, −8.358505979296918632239518392800, −7.61848870762004193638309186711, −6.95638146741475340569582445723, −5.86120657998933401256717519917, −5.11430793144295736663953593583, −4.09854389466643786070253476477, −3.25661912312057055898711123454, −2.26065795162526057830993395440, −1.06458588106369142400126897748,
1.06458588106369142400126897748, 2.26065795162526057830993395440, 3.25661912312057055898711123454, 4.09854389466643786070253476477, 5.11430793144295736663953593583, 5.86120657998933401256717519917, 6.95638146741475340569582445723, 7.61848870762004193638309186711, 8.358505979296918632239518392800, 8.997461625279297363034935811227