| L(s) = 1 | + 1.93·3-s − 1.24·7-s + 0.747·9-s − 0.513·11-s − 6.15·13-s + 4.51·17-s − 19-s − 2.41·21-s + 5.86·23-s − 4.36·27-s + 6.62·29-s + 6.41·31-s − 0.994·33-s + 1.40·37-s − 11.9·39-s + 10.6·41-s + 3.04·43-s + 1.99·47-s − 5.44·49-s + 8.74·51-s + 14.0·53-s − 1.93·57-s − 4.34·59-s + 10.7·61-s − 0.932·63-s + 9.89·67-s + 11.3·69-s + ⋯ |
| L(s) = 1 | + 1.11·3-s − 0.471·7-s + 0.249·9-s − 0.154·11-s − 1.70·13-s + 1.09·17-s − 0.229·19-s − 0.526·21-s + 1.22·23-s − 0.839·27-s + 1.23·29-s + 1.15·31-s − 0.173·33-s + 0.231·37-s − 1.90·39-s + 1.66·41-s + 0.464·43-s + 0.290·47-s − 0.777·49-s + 1.22·51-s + 1.93·53-s − 0.256·57-s − 0.565·59-s + 1.37·61-s − 0.117·63-s + 1.20·67-s + 1.36·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.489974997\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.489974997\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 1.93T + 3T^{2} \) |
| 7 | \( 1 + 1.24T + 7T^{2} \) |
| 11 | \( 1 + 0.513T + 11T^{2} \) |
| 13 | \( 1 + 6.15T + 13T^{2} \) |
| 17 | \( 1 - 4.51T + 17T^{2} \) |
| 23 | \( 1 - 5.86T + 23T^{2} \) |
| 29 | \( 1 - 6.62T + 29T^{2} \) |
| 31 | \( 1 - 6.41T + 31T^{2} \) |
| 37 | \( 1 - 1.40T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 - 3.04T + 43T^{2} \) |
| 47 | \( 1 - 1.99T + 47T^{2} \) |
| 53 | \( 1 - 14.0T + 53T^{2} \) |
| 59 | \( 1 + 4.34T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 - 9.89T + 67T^{2} \) |
| 71 | \( 1 - 7.42T + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 - 2.56T + 79T^{2} \) |
| 83 | \( 1 - 7.50T + 83T^{2} \) |
| 89 | \( 1 + 7.85T + 89T^{2} \) |
| 97 | \( 1 + 6.74T + 97T^{2} \) |
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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
−8.523013777818940021717068522527, −7.74631570886238804258717937599, −7.28406177307473383044538228024, −6.40041945442112974218599089878, −5.41321279029075149395497257224, −4.66297613549990937181686433619, −3.68736162001612522648546472761, −2.71447115400875784857741886769, −2.49775783671414909583897896506, −0.856980305515190160249224023033,
0.856980305515190160249224023033, 2.49775783671414909583897896506, 2.71447115400875784857741886769, 3.68736162001612522648546472761, 4.66297613549990937181686433619, 5.41321279029075149395497257224, 6.40041945442112974218599089878, 7.28406177307473383044538228024, 7.74631570886238804258717937599, 8.523013777818940021717068522527
