L(s) = 1 | − 2-s − 3-s − 4-s − 5-s + 6-s + 7-s + 3·8-s + 9-s + 10-s − 2·11-s + 12-s − 4·13-s − 14-s + 15-s − 16-s − 18-s + 4·19-s + 20-s − 21-s + 2·22-s − 4·23-s − 3·24-s + 25-s + 4·26-s − 27-s − 28-s − 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s + 0.288·12-s − 1.10·13-s − 0.267·14-s + 0.258·15-s − 1/4·16-s − 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.218·21-s + 0.426·22-s − 0.834·23-s − 0.612·24-s + 1/5·25-s + 0.784·26-s − 0.192·27-s − 0.188·28-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6927513927\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6927513927\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.16215695784990, −14.62209755392767, −14.06596688315274, −13.54925941842990, −12.89314383365720, −12.47964177621722, −11.68679364431894, −11.52663943334764, −10.68004795660244, −10.24152876815645, −9.728999006290565, −9.330882861261857, −8.505613862855372, −7.906578626550637, −7.677358658873247, −7.052790588886950, −6.297604774582971, −5.325057652643775, −5.168672051712994, −4.347847661555972, −3.949815086996624, −2.855754290246862, −2.132228292683614, −1.103697323148578, −0.4438972129497017,
0.4438972129497017, 1.103697323148578, 2.132228292683614, 2.855754290246862, 3.949815086996624, 4.347847661555972, 5.168672051712994, 5.325057652643775, 6.297604774582971, 7.052790588886950, 7.677358658873247, 7.906578626550637, 8.505613862855372, 9.330882861261857, 9.728999006290565, 10.24152876815645, 10.68004795660244, 11.52663943334764, 11.68679364431894, 12.47964177621722, 12.89314383365720, 13.54925941842990, 14.06596688315274, 14.62209755392767, 15.16215695784990