L(s) = 1 | − 3-s − 2·5-s + 9-s + 2·13-s + 2·15-s − 17-s + 4·19-s − 25-s − 27-s + 10·29-s − 8·31-s − 2·37-s − 2·39-s − 10·41-s + 12·43-s − 2·45-s − 7·49-s + 51-s + 6·53-s − 4·57-s − 12·59-s + 10·61-s − 4·65-s + 12·67-s − 10·73-s + 75-s − 8·79-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.554·13-s + 0.516·15-s − 0.242·17-s + 0.917·19-s − 1/5·25-s − 0.192·27-s + 1.85·29-s − 1.43·31-s − 0.328·37-s − 0.320·39-s − 1.56·41-s + 1.82·43-s − 0.298·45-s − 49-s + 0.140·51-s + 0.824·53-s − 0.529·57-s − 1.56·59-s + 1.28·61-s − 0.496·65-s + 1.46·67-s − 1.17·73-s + 0.115·75-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.00657061207413, −13.57566707504129, −12.81417860188555, −12.58474218816187, −11.90758918793706, −11.58515656895897, −11.23110202677195, −10.57307282799319, −10.25852669282105, −9.547389646840202, −9.083968624857401, −8.349900381580394, −8.136570999683867, −7.327070851354680, −7.060075290791609, −6.453281149708588, −5.794609842518784, −5.370435138971186, −4.671249372524654, −4.241672290218488, −3.551740270245009, −3.166291593306396, −2.276100325956477, −1.457185251893909, −0.7862592346574321, 0,
0.7862592346574321, 1.457185251893909, 2.276100325956477, 3.166291593306396, 3.551740270245009, 4.241672290218488, 4.671249372524654, 5.370435138971186, 5.794609842518784, 6.453281149708588, 7.060075290791609, 7.327070851354680, 8.136570999683867, 8.349900381580394, 9.083968624857401, 9.547389646840202, 10.25852669282105, 10.57307282799319, 11.23110202677195, 11.58515656895897, 11.90758918793706, 12.58474218816187, 12.81417860188555, 13.57566707504129, 14.00657061207413