| L(s) = 1 | + 2·3-s − 4·5-s + 9-s − 8·15-s + 2·17-s − 2·19-s + 8·23-s + 11·25-s − 4·27-s − 2·29-s − 4·31-s + 6·37-s + 2·41-s − 8·43-s − 4·45-s + 4·47-s + 4·51-s + 10·53-s − 4·57-s + 6·59-s + 4·61-s + 12·67-s + 16·69-s + 14·73-s + 22·75-s − 8·79-s − 11·81-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.78·5-s + 1/3·9-s − 2.06·15-s + 0.485·17-s − 0.458·19-s + 1.66·23-s + 11/5·25-s − 0.769·27-s − 0.371·29-s − 0.718·31-s + 0.986·37-s + 0.312·41-s − 1.21·43-s − 0.596·45-s + 0.583·47-s + 0.560·51-s + 1.37·53-s − 0.529·57-s + 0.781·59-s + 0.512·61-s + 1.46·67-s + 1.92·69-s + 1.63·73-s + 2.54·75-s − 0.900·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.803057761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.803057761\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.559490981278951083667367899441, −8.042828540799444591821513135257, −7.36451299174207798675058766123, −6.84990430480752226941204244096, −5.49837504030405723028581878340, −4.55788212286674073136007467732, −3.72000805486054490408222442872, −3.28169993806251707037692450068, −2.32564999119831306863506050617, −0.75804273115050893470668383393,
0.75804273115050893470668383393, 2.32564999119831306863506050617, 3.28169993806251707037692450068, 3.72000805486054490408222442872, 4.55788212286674073136007467732, 5.49837504030405723028581878340, 6.84990430480752226941204244096, 7.36451299174207798675058766123, 8.042828540799444591821513135257, 8.559490981278951083667367899441
