L(s) = 1 | − 2·3-s + 5-s + 9-s − 4·11-s + 13-s − 2·15-s + 6·17-s + 19-s − 23-s − 4·25-s + 4·27-s + 3·29-s − 7·31-s + 8·33-s − 10·37-s − 2·39-s + 10·41-s + 7·43-s + 45-s − 9·47-s − 12·51-s + 3·53-s − 4·55-s − 2·57-s − 6·61-s + 65-s + 6·67-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s + 0.277·13-s − 0.516·15-s + 1.45·17-s + 0.229·19-s − 0.208·23-s − 4/5·25-s + 0.769·27-s + 0.557·29-s − 1.25·31-s + 1.39·33-s − 1.64·37-s − 0.320·39-s + 1.56·41-s + 1.06·43-s + 0.149·45-s − 1.31·47-s − 1.68·51-s + 0.412·53-s − 0.539·55-s − 0.264·57-s − 0.768·61-s + 0.124·65-s + 0.733·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−16.84222870730652, −16.34791516233928, −15.97728025728122, −15.36792432642280, −14.47960528608312, −14.06209517046287, −13.40358041720057, −12.68696582829918, −12.27858692122648, −11.73303760490699, −10.87143668156903, −10.68503610405142, −9.964112233474118, −9.435679612375518, −8.541006567610347, −7.808749576586262, −7.332323803351710, −6.400403806015437, −5.791366265028502, −5.403795176101204, −4.879680057784072, −3.804393465898075, −3.014968248888953, −2.065728145863213, −1.033984887808569, 0,
1.033984887808569, 2.065728145863213, 3.014968248888953, 3.804393465898075, 4.879680057784072, 5.403795176101204, 5.791366265028502, 6.400403806015437, 7.332323803351710, 7.808749576586262, 8.541006567610347, 9.435679612375518, 9.964112233474118, 10.68503610405142, 10.87143668156903, 11.73303760490699, 12.27858692122648, 12.68696582829918, 13.40358041720057, 14.06209517046287, 14.47960528608312, 15.36792432642280, 15.97728025728122, 16.34791516233928, 16.84222870730652