| L(s) = 1 | − 2·7-s + 11-s + 2·17-s − 3·19-s − 4·23-s − 3·29-s + 31-s + 10·37-s + 41-s + 10·43-s + 10·47-s − 3·49-s − 6·53-s − 13·59-s − 10·61-s + 8·67-s + 9·71-s − 14·73-s − 2·77-s − 10·83-s + 13·89-s − 10·97-s − 101-s − 4·103-s + 12·107-s − 17·109-s − 6·113-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.301·11-s + 0.485·17-s − 0.688·19-s − 0.834·23-s − 0.557·29-s + 0.179·31-s + 1.64·37-s + 0.156·41-s + 1.52·43-s + 1.45·47-s − 3/7·49-s − 0.824·53-s − 1.69·59-s − 1.28·61-s + 0.977·67-s + 1.06·71-s − 1.63·73-s − 0.227·77-s − 1.09·83-s + 1.37·89-s − 1.01·97-s − 0.0995·101-s − 0.394·103-s + 1.16·107-s − 1.62·109-s − 0.564·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 13 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.64975376697863732710827277994, −6.65449791147337713789140623957, −6.12411406022150459113411978006, −5.59281945476441700387105976457, −4.46739597892129618440032459348, −3.98059111643115624168486050713, −3.06742772263835757636696095387, −2.32632413108316286099960471909, −1.21208192563995362424131490610, 0,
1.21208192563995362424131490610, 2.32632413108316286099960471909, 3.06742772263835757636696095387, 3.98059111643115624168486050713, 4.46739597892129618440032459348, 5.59281945476441700387105976457, 6.12411406022150459113411978006, 6.65449791147337713789140623957, 7.64975376697863732710827277994
