| L(s) = 1 | + 3-s + 5-s − 7-s + 9-s − 2·13-s + 15-s − 6·17-s − 4·19-s − 21-s + 25-s + 27-s + 6·29-s + 4·31-s − 35-s − 2·37-s − 2·39-s + 6·41-s + 8·43-s + 45-s + 12·47-s + 49-s − 6·51-s − 6·53-s − 4·57-s − 12·59-s − 2·61-s − 63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.554·13-s + 0.258·15-s − 1.45·17-s − 0.917·19-s − 0.218·21-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.718·31-s − 0.169·35-s − 0.328·37-s − 0.320·39-s + 0.937·41-s + 1.21·43-s + 0.149·45-s + 1.75·47-s + 1/7·49-s − 0.840·51-s − 0.824·53-s − 0.529·57-s − 1.56·59-s − 0.256·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.324935714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.324935714\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−7.978515145485646069187735022963, −7.33108854902409788194436714013, −6.41768772317016312336259897203, −6.20158409364579029263712797634, −4.92743542349778859885880619567, −4.44325577330637420989494109531, −3.55247748731974953241856335915, −2.48439822600300826951031607624, −2.18567106423834991001318834218, −0.73883825417033453364744445337,
0.73883825417033453364744445337, 2.18567106423834991001318834218, 2.48439822600300826951031607624, 3.55247748731974953241856335915, 4.44325577330637420989494109531, 4.92743542349778859885880619567, 6.20158409364579029263712797634, 6.41768772317016312336259897203, 7.33108854902409788194436714013, 7.978515145485646069187735022963
