| L(s) = 1 | − 19.1·5-s − 35.1·7-s + 26·11-s − 13·13-s + 36.2·17-s + 95.5·19-s − 161.·23-s + 240.·25-s + 91.3·29-s + 266.·31-s + 671.·35-s − 149.·37-s + 77.8·41-s − 183.·43-s − 60.6·47-s + 890.·49-s − 281.·53-s − 496.·55-s − 542.·59-s + 65.0·61-s + 248.·65-s + 1.03e3·67-s + 1.04e3·71-s + 483.·73-s − 912.·77-s + 1.33e3·79-s + 812.·83-s + ⋯ |
| L(s) = 1 | − 1.70·5-s − 1.89·7-s + 0.712·11-s − 0.277·13-s + 0.516·17-s + 1.15·19-s − 1.46·23-s + 1.92·25-s + 0.585·29-s + 1.54·31-s + 3.24·35-s − 0.664·37-s + 0.296·41-s − 0.651·43-s − 0.188·47-s + 2.59·49-s − 0.729·53-s − 1.21·55-s − 1.19·59-s + 0.136·61-s + 0.474·65-s + 1.88·67-s + 1.74·71-s + 0.774·73-s − 1.35·77-s + 1.90·79-s + 1.07·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 5 | \( 1 + 19.1T + 125T^{2} \) |
| 7 | \( 1 + 35.1T + 343T^{2} \) |
| 11 | \( 1 - 26T + 1.33e3T^{2} \) |
| 17 | \( 1 - 36.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 95.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 161.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 91.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 266.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 149.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 77.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 183.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 60.6T + 1.03e5T^{2} \) |
| 53 | \( 1 + 281.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 542.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 65.0T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.03e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.04e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 483.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.33e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 812.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 936.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 954.T + 9.12e5T^{2} \) |
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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
−8.281070193100072734471404005118, −7.78710342068889981386247026438, −6.77124169218390644567232598628, −6.42864839786140833994736941152, −5.16319157197850804973390641030, −3.99440462185048265865660132166, −3.55414415699425161581413133450, −2.78038519231245257871463135579, −0.884414296797356043782937990923, 0,
0.884414296797356043782937990923, 2.78038519231245257871463135579, 3.55414415699425161581413133450, 3.99440462185048265865660132166, 5.16319157197850804973390641030, 6.42864839786140833994736941152, 6.77124169218390644567232598628, 7.78710342068889981386247026438, 8.281070193100072734471404005118
