L(s) = 1 | + 2·5-s − 7-s − 3·9-s − 4·13-s − 4·17-s − 2·19-s − 23-s − 25-s − 6·29-s + 2·31-s − 2·35-s − 6·37-s − 2·41-s + 4·43-s − 6·45-s + 6·47-s + 49-s + 6·53-s − 2·61-s + 3·63-s − 8·65-s − 4·67-s + 12·71-s − 10·73-s − 8·79-s + 9·81-s − 6·83-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 0.377·7-s − 9-s − 1.10·13-s − 0.970·17-s − 0.458·19-s − 0.208·23-s − 1/5·25-s − 1.11·29-s + 0.359·31-s − 0.338·35-s − 0.986·37-s − 0.312·41-s + 0.609·43-s − 0.894·45-s + 0.875·47-s + 1/7·49-s + 0.824·53-s − 0.256·61-s + 0.377·63-s − 0.992·65-s − 0.488·67-s + 1.42·71-s − 1.17·73-s − 0.900·79-s + 81-s − 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−9.231996678754966744210536994798, −8.671639361492962144374602276344, −7.57393480776431837152329793829, −6.68582676046993793928713147131, −5.87132009664523065840617166846, −5.20410698425833403882159757730, −4.05081808929513917111357558171, −2.75675360438275398007849394382, −2.02121712693796991255226922987, 0,
2.02121712693796991255226922987, 2.75675360438275398007849394382, 4.05081808929513917111357558171, 5.20410698425833403882159757730, 5.87132009664523065840617166846, 6.68582676046993793928713147131, 7.57393480776431837152329793829, 8.671639361492962144374602276344, 9.231996678754966744210536994798