| L(s) = 1 | − 7-s + 5·13-s + 3·17-s − 2·19-s + 9·23-s − 5·25-s − 3·29-s − 5·31-s + 2·37-s − 6·41-s + 43-s + 6·47-s + 49-s + 3·53-s + 3·59-s − 10·61-s + 13·67-s − 9·71-s + 2·73-s + 10·79-s + 12·83-s + 15·89-s − 5·91-s + 8·97-s + 18·101-s + 13·103-s + 12·107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 1.38·13-s + 0.727·17-s − 0.458·19-s + 1.87·23-s − 25-s − 0.557·29-s − 0.898·31-s + 0.328·37-s − 0.937·41-s + 0.152·43-s + 0.875·47-s + 1/7·49-s + 0.412·53-s + 0.390·59-s − 1.28·61-s + 1.58·67-s − 1.06·71-s + 0.234·73-s + 1.12·79-s + 1.31·83-s + 1.58·89-s − 0.524·91-s + 0.812·97-s + 1.79·101-s + 1.28·103-s + 1.16·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.873892735\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.873892735\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 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
−8.891978601849373046435074140087, −7.949595272551558511484925825699, −7.24729027002187867605972756035, −6.38983991712328031863989081821, −5.75498451669473966849176010933, −4.92409316048517899781293445732, −3.77842535731702409100199368470, −3.29476397189543990973285365054, −2.01879358291610012679245055865, −0.856193565353031792325176418947,
0.856193565353031792325176418947, 2.01879358291610012679245055865, 3.29476397189543990973285365054, 3.77842535731702409100199368470, 4.92409316048517899781293445732, 5.75498451669473966849176010933, 6.38983991712328031863989081821, 7.24729027002187867605972756035, 7.949595272551558511484925825699, 8.891978601849373046435074140087
