| L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 14-s + 16-s − 20-s + 25-s − 28-s − 32-s + 35-s + 40-s + 49-s − 50-s + 2·53-s + 56-s + 2·59-s + 64-s − 70-s + 2·73-s − 2·79-s − 80-s − 2·97-s − 98-s + 100-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 14-s + 16-s − 20-s + 25-s − 28-s − 32-s + 35-s + 40-s + 49-s − 50-s + 2·53-s + 56-s + 2·59-s + 64-s − 70-s + 2·73-s − 2·79-s − 80-s − 2·97-s − 98-s + 100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5042071276\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5042071276\) |
| \(L(1)\) |
|
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 | \( 1 + T \) |
| good | 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( ( 1 - T )^{2} \) |
| 59 | \( ( 1 - T )^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( ( 1 - T )^{2} \) |
| 79 | \( ( 1 + T )^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 + T )^{2} \) |
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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
−8.907552963482613960293790935735, −8.511372582747375206941125997049, −7.54429669642036282243439717436, −7.03726282087269493369091206194, −6.30645512637126622857685204113, −5.35983874228518724166298548057, −4.03368261583746502863354012564, −3.29370984578126772749406503855, −2.35052102646449820430057829549, −0.74781351685986104441415166383,
0.74781351685986104441415166383, 2.35052102646449820430057829549, 3.29370984578126772749406503855, 4.03368261583746502863354012564, 5.35983874228518724166298548057, 6.30645512637126622857685204113, 7.03726282087269493369091206194, 7.54429669642036282243439717436, 8.511372582747375206941125997049, 8.907552963482613960293790935735
