L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s + 11-s − 12-s + 6·13-s + 14-s + 15-s + 16-s + 2·17-s − 18-s − 20-s + 21-s − 22-s + 24-s + 25-s − 6·26-s − 27-s − 28-s − 6·29-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 − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s + 1.66·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.223·20-s + 0.218·21-s − 0.213·22-s + 0.204·24-s + 1/5·25-s − 1.17·26-s − 0.192·27-s − 0.188·28-s − 1.11·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9049802923\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9049802923\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−9.103626326000292140361845001139, −8.196945598193707075032817184973, −7.57420452246541303205849993325, −6.63646672081543735783708127570, −6.08598496870986425748720894271, −5.23471860520139562290057471514, −3.93137101371456974009037000769, −3.36110416656923312558321095025, −1.82137842479735763109671871262, −0.72006291408836666650207109861,
0.72006291408836666650207109861, 1.82137842479735763109671871262, 3.36110416656923312558321095025, 3.93137101371456974009037000769, 5.23471860520139562290057471514, 6.08598496870986425748720894271, 6.63646672081543735783708127570, 7.57420452246541303205849993325, 8.196945598193707075032817184973, 9.103626326000292140361845001139