| L(s) = 1 | − 5-s − 4·11-s − 6·13-s + 6·17-s − 4·19-s + 4·23-s + 25-s − 2·29-s − 8·31-s − 6·37-s + 6·41-s − 8·43-s + 6·53-s + 4·55-s + 4·59-s + 10·61-s + 6·65-s − 8·67-s − 12·71-s + 14·73-s − 16·79-s + 12·83-s − 6·85-s + 14·89-s + 4·95-s − 18·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.20·11-s − 1.66·13-s + 1.45·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s − 0.371·29-s − 1.43·31-s − 0.986·37-s + 0.937·41-s − 1.21·43-s + 0.824·53-s + 0.539·55-s + 0.520·59-s + 1.28·61-s + 0.744·65-s − 0.977·67-s − 1.42·71-s + 1.63·73-s − 1.80·79-s + 1.31·83-s − 0.650·85-s + 1.48·89-s + 0.410·95-s − 1.82·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4164328074\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4164328074\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−13.27815180039510, −12.80699641142487, −12.52299456481590, −12.02771462102573, −11.57853063982218, −10.89096731520557, −10.54707214754762, −10.04047029101987, −9.678508810351579, −8.996904934790834, −8.524918537673979, −7.906671381172504, −7.521089008787415, −7.164131804627830, −6.663507215001522, −5.714941864733839, −5.367476963717175, −4.991870706025922, −4.379172911331507, −3.647092837893990, −3.177435611934320, −2.498948118455654, −2.068062840878164, −1.152846790359828, −0.1989654319283933,
0.1989654319283933, 1.152846790359828, 2.068062840878164, 2.498948118455654, 3.177435611934320, 3.647092837893990, 4.379172911331507, 4.991870706025922, 5.367476963717175, 5.714941864733839, 6.663507215001522, 7.164131804627830, 7.521089008787415, 7.906671381172504, 8.524918537673979, 8.996904934790834, 9.678508810351579, 10.04047029101987, 10.54707214754762, 10.89096731520557, 11.57853063982218, 12.02771462102573, 12.52299456481590, 12.80699641142487, 13.27815180039510
