L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s + 4·11-s + 12-s + 2·13-s − 15-s + 16-s + 17-s + 18-s − 4·19-s − 20-s + 4·22-s + 4·23-s + 24-s + 25-s + 2·26-s + 27-s + 2·29-s − 30-s − 4·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s + 0.288·12-s + 0.554·13-s − 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.917·19-s − 0.223·20-s + 0.852·22-s + 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.371·29-s − 0.182·30-s − 0.718·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.585843607\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.585843607\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−11.07946823169159140727754934715, −10.10134613511098837605690756967, −8.939698841198500618925607417374, −8.297752648192488937179725745614, −7.06236087576745542321631935950, −6.43028480692664490516364270008, −5.06382131673478537430485419157, −3.97578701568053985951665093752, −3.24401384153019634960819626503, −1.64973784957677039929822316285,
1.64973784957677039929822316285, 3.24401384153019634960819626503, 3.97578701568053985951665093752, 5.06382131673478537430485419157, 6.43028480692664490516364270008, 7.06236087576745542321631935950, 8.297752648192488937179725745614, 8.939698841198500618925607417374, 10.10134613511098837605690756967, 11.07946823169159140727754934715