L(s) = 1 | + 2-s + 4-s − 5-s + 3·7-s + 8-s − 10-s + 5·11-s − 2·13-s + 3·14-s + 16-s + 7·17-s − 2·19-s − 20-s + 5·22-s − 4·23-s + 25-s − 2·26-s + 3·28-s + 5·29-s − 7·31-s + 32-s + 7·34-s − 3·35-s + 37-s − 2·38-s − 40-s − 41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.13·7-s + 0.353·8-s − 0.316·10-s + 1.50·11-s − 0.554·13-s + 0.801·14-s + 1/4·16-s + 1.69·17-s − 0.458·19-s − 0.223·20-s + 1.06·22-s − 0.834·23-s + 1/5·25-s − 0.392·26-s + 0.566·28-s + 0.928·29-s − 1.25·31-s + 0.176·32-s + 1.20·34-s − 0.507·35-s + 0.164·37-s − 0.324·38-s − 0.158·40-s − 0.156·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.562837601\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.562837601\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 - T \) |
good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−8.479363894691396823198428766432, −7.73463508398268072117254848694, −7.22405775820118106566112790586, −6.24681831042771853874863288742, −5.54105138781531173276253848284, −4.65945553858662436400984934494, −4.05434382945432804672464758693, −3.27535295134737187498488023443, −2.02898702699814251887552526958, −1.11392896224171524842174576504,
1.11392896224171524842174576504, 2.02898702699814251887552526958, 3.27535295134737187498488023443, 4.05434382945432804672464758693, 4.65945553858662436400984934494, 5.54105138781531173276253848284, 6.24681831042771853874863288742, 7.22405775820118106566112790586, 7.73463508398268072117254848694, 8.479363894691396823198428766432