| L(s) = 1 | − 2-s − 4-s + 3·8-s − 3·9-s + 4·11-s − 16-s + 3·18-s − 4·22-s − 8·23-s + 2·29-s − 5·32-s + 3·36-s + 6·37-s + 12·43-s − 4·44-s + 8·46-s + 10·53-s − 2·58-s + 7·64-s − 4·67-s + 16·71-s − 9·72-s − 6·74-s + 8·79-s + 9·81-s − 12·86-s + 12·88-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s − 9-s + 1.20·11-s − 1/4·16-s + 0.707·18-s − 0.852·22-s − 1.66·23-s + 0.371·29-s − 0.883·32-s + 1/2·36-s + 0.986·37-s + 1.82·43-s − 0.603·44-s + 1.17·46-s + 1.37·53-s − 0.262·58-s + 7/8·64-s − 0.488·67-s + 1.89·71-s − 1.06·72-s − 0.697·74-s + 0.900·79-s + 81-s − 1.29·86-s + 1.27·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8646032790\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8646032790\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−9.540102098470066783229159054034, −8.959096240229079857657424888696, −8.258764783821744494006777469815, −7.53770562837158976453911852227, −6.38920763142189327962156871867, −5.62983515568003563476629870364, −4.44056415760512758846670567036, −3.68334690525640900028863674377, −2.20922818309632113036387473157, −0.77987991622378332753572806223,
0.77987991622378332753572806223, 2.20922818309632113036387473157, 3.68334690525640900028863674377, 4.44056415760512758846670567036, 5.62983515568003563476629870364, 6.38920763142189327962156871867, 7.53770562837158976453911852227, 8.258764783821744494006777469815, 8.959096240229079857657424888696, 9.540102098470066783229159054034
