| L(s) = 1 | − 5-s + 5·11-s + 3·13-s − 17-s − 6·19-s − 6·23-s + 25-s + 9·29-s + 4·31-s + 2·37-s − 4·41-s + 10·43-s − 47-s − 4·53-s − 5·55-s − 8·59-s + 8·61-s − 3·65-s + 12·67-s − 8·71-s − 2·73-s + 13·79-s − 4·83-s + 85-s + 4·89-s + 6·95-s + 13·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.50·11-s + 0.832·13-s − 0.242·17-s − 1.37·19-s − 1.25·23-s + 1/5·25-s + 1.67·29-s + 0.718·31-s + 0.328·37-s − 0.624·41-s + 1.52·43-s − 0.145·47-s − 0.549·53-s − 0.674·55-s − 1.04·59-s + 1.02·61-s − 0.372·65-s + 1.46·67-s − 0.949·71-s − 0.234·73-s + 1.46·79-s − 0.439·83-s + 0.108·85-s + 0.423·89-s + 0.615·95-s + 1.31·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.036642799\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.036642799\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−7.952707246904741266133813893709, −6.85878606595018715736523696816, −6.40811238969892672963700075067, −5.97339475257182914414081076593, −4.74727235810471928622849127839, −4.15296734724503820777886273119, −3.69478376899115160566499610240, −2.63882627312324206303480403669, −1.67251185109195743147827319112, −0.71878972479515514816788377195,
0.71878972479515514816788377195, 1.67251185109195743147827319112, 2.63882627312324206303480403669, 3.69478376899115160566499610240, 4.15296734724503820777886273119, 4.74727235810471928622849127839, 5.97339475257182914414081076593, 6.40811238969892672963700075067, 6.85878606595018715736523696816, 7.952707246904741266133813893709
