| L(s) = 1 | + 46·3-s − 1.03e4·7-s − 1.75e4·9-s + 5.56e3·11-s − 4.59e4·13-s + 3.81e5·17-s − 6.10e5·19-s − 4.74e5·21-s − 1.44e6·23-s − 1.71e6·27-s + 5.38e6·29-s − 3.05e6·31-s + 2.56e5·33-s − 1.28e7·37-s − 2.11e6·39-s − 3.37e7·41-s − 3.68e7·43-s − 4.41e7·47-s + 6.61e7·49-s + 1.75e7·51-s − 2.97e7·53-s − 2.80e7·57-s + 6.55e7·59-s + 4.01e7·61-s + 1.81e8·63-s − 1.15e8·67-s − 6.66e7·69-s + ⋯ |
| L(s) = 1 | + 0.327·3-s − 1.62·7-s − 0.892·9-s + 0.114·11-s − 0.446·13-s + 1.10·17-s − 1.07·19-s − 0.532·21-s − 1.07·23-s − 0.620·27-s + 1.41·29-s − 0.593·31-s + 0.0375·33-s − 1.13·37-s − 0.146·39-s − 1.86·41-s − 1.64·43-s − 1.32·47-s + 1.63·49-s + 0.363·51-s − 0.517·53-s − 0.352·57-s + 0.704·59-s + 0.371·61-s + 1.44·63-s − 0.701·67-s − 0.353·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.5745200031\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5745200031\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 46 T + p^{9} T^{2} \) |
| 7 | \( 1 + 1474 p T + p^{9} T^{2} \) |
| 11 | \( 1 - 5568 T + p^{9} T^{2} \) |
| 13 | \( 1 + 45986 T + p^{9} T^{2} \) |
| 17 | \( 1 - 381318 T + p^{9} T^{2} \) |
| 19 | \( 1 + 610460 T + p^{9} T^{2} \) |
| 23 | \( 1 + 1447914 T + p^{9} T^{2} \) |
| 29 | \( 1 - 5385510 T + p^{9} T^{2} \) |
| 31 | \( 1 + 3053852 T + p^{9} T^{2} \) |
| 37 | \( 1 + 12889442 T + p^{9} T^{2} \) |
| 41 | \( 1 + 33786618 T + p^{9} T^{2} \) |
| 43 | \( 1 + 36886234 T + p^{9} T^{2} \) |
| 47 | \( 1 + 44163798 T + p^{9} T^{2} \) |
| 53 | \( 1 + 29746266 T + p^{9} T^{2} \) |
| 59 | \( 1 - 65575380 T + p^{9} T^{2} \) |
| 61 | \( 1 - 40183202 T + p^{9} T^{2} \) |
| 67 | \( 1 + 115706158 T + p^{9} T^{2} \) |
| 71 | \( 1 - 231681708 T + p^{9} T^{2} \) |
| 73 | \( 1 + 358691906 T + p^{9} T^{2} \) |
| 79 | \( 1 - 486017080 T + p^{9} T^{2} \) |
| 83 | \( 1 - 251168886 T + p^{9} T^{2} \) |
| 89 | \( 1 + 526039110 T + p^{9} T^{2} \) |
| 97 | \( 1 - 1075981438 T + p^{9} T^{2} \) |
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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
−9.912240990131195365161947093419, −8.813375698463358121120457518124, −8.100130276190783692869721438786, −6.80037080359821038778589486664, −6.17954167848128510929514364520, −5.09290515900693812217626328262, −3.60058048770624251966805666941, −3.07944997431985410324836597794, −1.92116646205910604334400000375, −0.28993492119979290824994227584,
0.28993492119979290824994227584, 1.92116646205910604334400000375, 3.07944997431985410324836597794, 3.60058048770624251966805666941, 5.09290515900693812217626328262, 6.17954167848128510929514364520, 6.80037080359821038778589486664, 8.100130276190783692869721438786, 8.813375698463358121120457518124, 9.912240990131195365161947093419
