| L(s) = 1 | + 5·7-s − 5·13-s + 19-s + 7·31-s + 10·37-s + 5·43-s + 18·49-s − 13·61-s + 5·67-s + 10·73-s + 4·79-s − 25·91-s − 5·97-s + 20·103-s − 19·109-s + ⋯ |
| L(s) = 1 | + 1.88·7-s − 1.38·13-s + 0.229·19-s + 1.25·31-s + 1.64·37-s + 0.762·43-s + 18/7·49-s − 1.66·61-s + 0.610·67-s + 1.17·73-s + 0.450·79-s − 2.62·91-s − 0.507·97-s + 1.97·103-s − 1.81·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.339986645\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.339986645\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−8.370230055573889608400155652183, −7.79418116555211414161142515552, −7.35392867503457223264572962479, −6.29199830790675937341358085541, −5.34239834622774018107892237126, −4.74317936109230330427568877327, −4.20473686362829678532429832272, −2.78328703779702579692296854925, −2.03300802883451810561552681476, −0.935279568632096924129002915552,
0.935279568632096924129002915552, 2.03300802883451810561552681476, 2.78328703779702579692296854925, 4.20473686362829678532429832272, 4.74317936109230330427568877327, 5.34239834622774018107892237126, 6.29199830790675937341358085541, 7.35392867503457223264572962479, 7.79418116555211414161142515552, 8.370230055573889608400155652183
