L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s + 11-s − 6·13-s − 14-s + 16-s + 2·17-s − 20-s + 22-s − 8·23-s + 25-s − 6·26-s − 28-s + 6·29-s − 8·31-s + 32-s + 2·34-s + 35-s + 10·37-s − 40-s + 10·41-s − 4·43-s + 44-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s + 0.301·11-s − 1.66·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.223·20-s + 0.213·22-s − 1.66·23-s + 1/5·25-s − 1.17·26-s − 0.188·28-s + 1.11·29-s − 1.43·31-s + 0.176·32-s + 0.342·34-s + 0.169·35-s + 1.64·37-s − 0.158·40-s + 1.56·41-s − 0.609·43-s + 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.353605391\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.353605391\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 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 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−7.66926160554758258570765998912, −7.33922126249333767017822763113, −6.47933854273315555213845540585, −5.80268561137230341022673444775, −5.08646464484723051703128743978, −4.25429156945067898205571587686, −3.77443468645511037683174188113, −2.73026719740563098272093135163, −2.14689276550243545064801352569, −0.67753181162717795029178232970,
0.67753181162717795029178232970, 2.14689276550243545064801352569, 2.73026719740563098272093135163, 3.77443468645511037683174188113, 4.25429156945067898205571587686, 5.08646464484723051703128743978, 5.80268561137230341022673444775, 6.47933854273315555213845540585, 7.33922126249333767017822763113, 7.66926160554758258570765998912