L(s) = 1 | + 2·3-s − 5-s + 3·7-s + 9-s + 4·13-s − 2·15-s + 3·17-s + 19-s + 6·21-s + 8·23-s − 4·25-s − 4·27-s + 2·29-s + 4·31-s − 3·35-s + 10·37-s + 8·39-s − 10·41-s − 43-s − 45-s − 47-s + 2·49-s + 6·51-s − 4·53-s + 2·57-s + 6·59-s + 13·61-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 0.447·5-s + 1.13·7-s + 1/3·9-s + 1.10·13-s − 0.516·15-s + 0.727·17-s + 0.229·19-s + 1.30·21-s + 1.66·23-s − 4/5·25-s − 0.769·27-s + 0.371·29-s + 0.718·31-s − 0.507·35-s + 1.64·37-s + 1.28·39-s − 1.56·41-s − 0.152·43-s − 0.149·45-s − 0.145·47-s + 2/7·49-s + 0.840·51-s − 0.549·53-s + 0.264·57-s + 0.781·59-s + 1.66·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.935096026\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.935096026\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.933665480997825519415597461102, −7.32424896802954806677465404753, −6.44587861158160350440717785391, −5.57209357785375076961912739494, −4.85244736948764917088509248105, −4.08026045073955845688679172688, −3.36880239235033917183403851835, −2.77013505033142815042798931153, −1.74404941065737830362193213595, −0.971870007004528213384011971816,
0.971870007004528213384011971816, 1.74404941065737830362193213595, 2.77013505033142815042798931153, 3.36880239235033917183403851835, 4.08026045073955845688679172688, 4.85244736948764917088509248105, 5.57209357785375076961912739494, 6.44587861158160350440717785391, 7.32424896802954806677465404753, 7.933665480997825519415597461102