L(s) = 1 | + 2-s + 3-s − 4-s + 5-s + 6-s + 7-s − 3·8-s + 9-s + 10-s + 11-s − 12-s − 13-s + 14-s + 15-s − 16-s + 18-s − 20-s + 21-s + 22-s − 3·24-s + 25-s − 26-s + 27-s − 28-s − 6·29-s + 30-s + 7·31-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 + 0.301·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.258·15-s − 1/4·16-s + 0.235·18-s − 0.223·20-s + 0.218·21-s + 0.213·22-s − 0.612·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s − 1.11·29-s + 0.182·30-s + 1.25·31-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
\(L(1)\) |
\(\approx\) |
\(3.901441209\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.901441209\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−14.95676225847255, −14.47342804813787, −14.13340085942447, −13.55920226775931, −13.08121701499926, −12.73656847612059, −12.02742192515731, −11.55383310382519, −10.89654181596087, −10.15399598952737, −9.610470288942653, −9.184969836294670, −8.733081984189003, −7.910267037624410, −7.637638260814906, −6.664033436145199, −6.102362538611007, −5.600288584091826, −4.789356314806210, −4.418042305141774, −3.788625390498179, −2.999402134956419, −2.487568338256464, −1.589616118344409, −0.6665656962130901,
0.6665656962130901, 1.589616118344409, 2.487568338256464, 2.999402134956419, 3.788625390498179, 4.418042305141774, 4.789356314806210, 5.600288584091826, 6.102362538611007, 6.664033436145199, 7.637638260814906, 7.910267037624410, 8.733081984189003, 9.184969836294670, 9.610470288942653, 10.15399598952737, 10.89654181596087, 11.55383310382519, 12.02742192515731, 12.73656847612059, 13.08121701499926, 13.55920226775931, 14.13340085942447, 14.47342804813787, 14.95676225847255