L(s) = 1 | − 0.470·3-s − 5-s + 0.470·7-s − 2.77·9-s − 1.77·11-s − 0.941·13-s + 0.470·15-s + 4.49·17-s + 2.24·19-s − 0.221·21-s + 7.55·23-s + 25-s + 2.71·27-s − 2·29-s − 0.366·31-s + 0.837·33-s − 0.470·35-s − 37-s + 0.443·39-s − 6.83·41-s + 3.55·43-s + 2.77·45-s + 0.470·47-s − 6.77·49-s − 2.11·51-s − 12.2·53-s + 1.77·55-s + ⋯ |
L(s) = 1 | − 0.271·3-s − 0.447·5-s + 0.177·7-s − 0.926·9-s − 0.536·11-s − 0.261·13-s + 0.121·15-s + 1.09·17-s + 0.515·19-s − 0.0483·21-s + 1.57·23-s + 0.200·25-s + 0.523·27-s − 0.371·29-s − 0.0658·31-s + 0.145·33-s − 0.0795·35-s − 0.164·37-s + 0.0709·39-s − 1.06·41-s + 0.542·43-s + 0.414·45-s + 0.0686·47-s − 0.968·49-s − 0.296·51-s − 1.68·53-s + 0.239·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 + T \) |
good | 3 | \( 1 + 0.470T + 3T^{2} \) |
| 7 | \( 1 - 0.470T + 7T^{2} \) |
| 11 | \( 1 + 1.77T + 11T^{2} \) |
| 13 | \( 1 + 0.941T + 13T^{2} \) |
| 17 | \( 1 - 4.49T + 17T^{2} \) |
| 19 | \( 1 - 2.24T + 19T^{2} \) |
| 23 | \( 1 - 7.55T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 0.366T + 31T^{2} \) |
| 41 | \( 1 + 6.83T + 41T^{2} \) |
| 43 | \( 1 - 3.55T + 43T^{2} \) |
| 47 | \( 1 - 0.470T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 + 0.131T + 59T^{2} \) |
| 61 | \( 1 + 9.55T + 61T^{2} \) |
| 67 | \( 1 + 6.24T + 67T^{2} \) |
| 71 | \( 1 - 3.16T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 + 4.13T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 - 3.43T + 89T^{2} \) |
| 97 | \( 1 - 0.325T + 97T^{2} \) |
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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
−8.286816598390339473638099836970, −7.65927568508781114513831836888, −6.96735976068548905732362274243, −5.94700562548802613760262845723, −5.26358237035135394907698385457, −4.65311659788273141825230952043, −3.32946100355292609544473313751, −2.86428918245613172654219794597, −1.35951038978963472656589643141, 0,
1.35951038978963472656589643141, 2.86428918245613172654219794597, 3.32946100355292609544473313751, 4.65311659788273141825230952043, 5.26358237035135394907698385457, 5.94700562548802613760262845723, 6.96735976068548905732362274243, 7.65927568508781114513831836888, 8.286816598390339473638099836970