| L(s) = 1 | − 4·5-s − 3·9-s + 4·13-s − 8·17-s + 11·25-s + 10·29-s + 2·37-s − 8·41-s + 12·45-s + 14·53-s + 12·61-s − 16·65-s + 16·73-s + 9·81-s + 32·85-s + 16·89-s + 8·97-s − 20·101-s − 6·109-s − 14·113-s − 12·117-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 9-s + 1.10·13-s − 1.94·17-s + 11/5·25-s + 1.85·29-s + 0.328·37-s − 1.24·41-s + 1.78·45-s + 1.92·53-s + 1.53·61-s − 1.98·65-s + 1.87·73-s + 81-s + 3.47·85-s + 1.69·89-s + 0.812·97-s − 1.99·101-s − 0.574·109-s − 1.31·113-s − 1.10·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8616555467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8616555467\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−9.083201307753543769979419428054, −8.396621168864933987238022281800, −8.197509550250461170763039761778, −6.93218317388480289991978591057, −6.43989265358166028466611552196, −5.14312271521888697700030317914, −4.24873327227852142886149189756, −3.55887344410299009293086169964, −2.53443050487252517839586499541, −0.63247991375959544191760657111,
0.63247991375959544191760657111, 2.53443050487252517839586499541, 3.55887344410299009293086169964, 4.24873327227852142886149189756, 5.14312271521888697700030317914, 6.43989265358166028466611552196, 6.93218317388480289991978591057, 8.197509550250461170763039761778, 8.396621168864933987238022281800, 9.083201307753543769979419428054
