| L(s) = 1 | + 3-s − 4.13·5-s + 9-s + 2.29·11-s + 2.43·13-s − 4.13·15-s + 3.71·17-s + 7.84·19-s − 1.70·23-s + 12.0·25-s + 27-s − 3.25·29-s + 1.80·31-s + 2.29·33-s + 5.65·37-s + 2.43·39-s − 6.81·41-s − 5.44·43-s − 4.13·45-s + 13.2·47-s + 3.71·51-s − 9.09·53-s − 9.50·55-s + 7.84·57-s − 14.9·59-s − 2.16·61-s − 10.0·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.84·5-s + 0.333·9-s + 0.693·11-s + 0.674·13-s − 1.06·15-s + 0.900·17-s + 1.80·19-s − 0.354·23-s + 2.41·25-s + 0.192·27-s − 0.603·29-s + 0.324·31-s + 0.400·33-s + 0.929·37-s + 0.389·39-s − 1.06·41-s − 0.829·43-s − 0.616·45-s + 1.93·47-s + 0.519·51-s − 1.24·53-s − 1.28·55-s + 1.03·57-s − 1.94·59-s − 0.277·61-s − 1.24·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.051659251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.051659251\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 4.13T + 5T^{2} \) |
| 11 | \( 1 - 2.29T + 11T^{2} \) |
| 13 | \( 1 - 2.43T + 13T^{2} \) |
| 17 | \( 1 - 3.71T + 17T^{2} \) |
| 19 | \( 1 - 7.84T + 19T^{2} \) |
| 23 | \( 1 + 1.70T + 23T^{2} \) |
| 29 | \( 1 + 3.25T + 29T^{2} \) |
| 31 | \( 1 - 1.80T + 31T^{2} \) |
| 37 | \( 1 - 5.65T + 37T^{2} \) |
| 41 | \( 1 + 6.81T + 41T^{2} \) |
| 43 | \( 1 + 5.44T + 43T^{2} \) |
| 47 | \( 1 - 13.2T + 47T^{2} \) |
| 53 | \( 1 + 9.09T + 53T^{2} \) |
| 59 | \( 1 + 14.9T + 59T^{2} \) |
| 61 | \( 1 + 2.16T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 - 3.95T + 71T^{2} \) |
| 73 | \( 1 - 9.68T + 73T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−7.67127438328775633534313478761, −7.36823974797432707068421715429, −6.50632098018997179727011944704, −5.62461006369693460399004349373, −4.68324331258651842466753528589, −4.07968623477262768367888796387, −3.29217750645491835116996973254, −3.14679361859741723150044891959, −1.54579749207019016309165552653, −0.71670789242166631389331725933,
0.71670789242166631389331725933, 1.54579749207019016309165552653, 3.14679361859741723150044891959, 3.29217750645491835116996973254, 4.07968623477262768367888796387, 4.68324331258651842466753528589, 5.62461006369693460399004349373, 6.50632098018997179727011944704, 7.36823974797432707068421715429, 7.67127438328775633534313478761
