L(s) = 1 | − 2·5-s − 3·9-s − 11-s − 2·13-s + 3·17-s − 19-s + 3·23-s − 25-s + 6·29-s + 10·31-s + 4·37-s + 10·41-s − 4·43-s + 6·45-s − 13·47-s − 12·53-s + 2·55-s + 6·59-s + 9·61-s + 4·65-s + 14·67-s − 2·71-s − 73-s − 8·79-s + 9·81-s − 5·83-s − 6·85-s + ⋯ |
L(s) = 1 | − 0.894·5-s − 9-s − 0.301·11-s − 0.554·13-s + 0.727·17-s − 0.229·19-s + 0.625·23-s − 1/5·25-s + 1.11·29-s + 1.79·31-s + 0.657·37-s + 1.56·41-s − 0.609·43-s + 0.894·45-s − 1.89·47-s − 1.64·53-s + 0.269·55-s + 0.781·59-s + 1.15·61-s + 0.496·65-s + 1.71·67-s − 0.237·71-s − 0.117·73-s − 0.900·79-s + 81-s − 0.548·83-s − 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 9 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−7.83048266201333108811346303564, −6.81563977494501388317934833403, −6.24650399410782868912949070124, −5.33138863419463542404480332818, −4.74321414416839488068228377423, −3.94069100679646547916131711351, −3.00730814588054618134655184542, −2.56081696310821466864150706627, −1.07292356908452516128914765614, 0,
1.07292356908452516128914765614, 2.56081696310821466864150706627, 3.00730814588054618134655184542, 3.94069100679646547916131711351, 4.74321414416839488068228377423, 5.33138863419463542404480332818, 6.24650399410782868912949070124, 6.81563977494501388317934833403, 7.83048266201333108811346303564