| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 3·7-s + 8-s + 9-s + 2·11-s − 12-s + 2·13-s − 3·14-s + 16-s − 5·17-s + 18-s + 3·21-s + 2·22-s − 24-s − 5·25-s + 2·26-s − 27-s − 3·28-s + 3·29-s + 8·31-s + 32-s − 2·33-s − 5·34-s + 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.603·11-s − 0.288·12-s + 0.554·13-s − 0.801·14-s + 1/4·16-s − 1.21·17-s + 0.235·18-s + 0.654·21-s + 0.426·22-s − 0.204·24-s − 25-s + 0.392·26-s − 0.192·27-s − 0.566·28-s + 0.557·29-s + 1.43·31-s + 0.176·32-s − 0.348·33-s − 0.857·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.709378542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.709378542\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.64602491221523, −13.29317387430172, −12.98938653208752, −12.35155720327731, −11.79127322565511, −11.64590935383606, −10.87915917298125, −10.53748946587328, −9.904071060126869, −9.443512926048569, −8.939994120831885, −8.286276498868239, −7.626046828245549, −7.030117059467588, −6.404137093379530, −6.260883675207538, −5.813241197925590, −5.016334237111232, −4.317166759480621, −4.088770684767517, −3.390604837408132, −2.693493011345076, −2.166855672346991, −1.198133365883621, −0.5152338558327769,
0.5152338558327769, 1.198133365883621, 2.166855672346991, 2.693493011345076, 3.390604837408132, 4.088770684767517, 4.317166759480621, 5.016334237111232, 5.813241197925590, 6.260883675207538, 6.404137093379530, 7.030117059467588, 7.626046828245549, 8.286276498868239, 8.939994120831885, 9.443512926048569, 9.904071060126869, 10.53748946587328, 10.87915917298125, 11.64590935383606, 11.79127322565511, 12.35155720327731, 12.98938653208752, 13.29317387430172, 13.64602491221523
